Page 1 chap-02 B.V.Ramana August 30, 2006 10:15 Chapter2 Sequences and Series INTRODUCTION The study of convergence and divergence of a se- quence, which is an ordered list of things, is a prereq- uisit for in?nite series. The unit square in the ?gure can be expressed as an in?nite (geometric) series 1= 1 2 + 1 4 + 1 8 + 1 16 + 1 32 +··· Several functions can be expressed as “in?nite polynomials” (known as “powerser ies”) using the concept of in?nite series. By Fourier series, certain functions can be represented as an in?nite sum of trigonometric functions. Using in?nite series, differential equations in problems of signal transmission, chemical diffusion, vibration and heat ?ow can be solved and non elementary integrals evaluated. The in?nite process of summing of an in?nite series is a puzzle for centuries convergence and divergence of in?nite series plays an important role in engineering applications. 1 2 1 4 1 8 1 16 2.1 SEQUENCES A sequence is a function from the domain set of natural numbersN to any setS. Real sequence isafunctionfromN to R, the set of real numbers; denoted by f :N ? R. Thus the real sequence f is set of all ordered pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs (n,f (n)) with n a positive integer. Notation: Sincethedomainofasequenceisalways thesame(thesetofpositiveintegers)asequencemay be written as{f (n)} instead of{n,f (n)}. Examples: 1. n, 1 n = 1 n = 1, 1 2 , 1 3 , 1 4 ,... 1 n ... 2. n, 1 2 n-1 = 1 2 n-1 = 1, 1 2 , 1 2 2 , 1 2 3 , 1 2 4 ,..., 1 2 n-1 ,... constant sequence where range is singleton set{c}, c= constant. Example: {3, 3, 3, 3,...} Null sequence{0, 0, 0,..., 0,...} A sequence is also denoted by{a n } whose ordinate y = a n at the abscissax = n. Thus in a sequence for each positive integern, a numbera n is assigned and is denoted as a n or( a n )or {a n }={a(1),a(2),a(3),...,a(n),...} ={a 1 ,a 2 ,a 3 ,...,a n ,...} Herea 1 ,a 2 ,a 3 ,...a n , are known as the ?rst, second, third and nth terms of the sequence. In?nite sequence is a sequence in which the numberoftermsisin?nite,andisdenotedby {a n } 8 n=1 . Ontheotherhand,?nitesequencedenotedby{a n } m n=1 containsonlya?nitenumberofterms( m=?nite). Bounded sequence A sequence {a n } is said to be bounded if there exists numbers m and M such thatm<a n <M forevery n, otherwise it is said to be unbounded. Monotonic sequence A sequence{a n }issaidtobe a. monotonically increasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· 2.1 Page 2 chap-02 B.V.Ramana August 30, 2006 10:15 Chapter2 Sequences and Series INTRODUCTION The study of convergence and divergence of a se- quence, which is an ordered list of things, is a prereq- uisit for in?nite series. The unit square in the ?gure can be expressed as an in?nite (geometric) series 1= 1 2 + 1 4 + 1 8 + 1 16 + 1 32 +··· Several functions can be expressed as “in?nite polynomials” (known as “powerser ies”) using the concept of in?nite series. By Fourier series, certain functions can be represented as an in?nite sum of trigonometric functions. Using in?nite series, differential equations in problems of signal transmission, chemical diffusion, vibration and heat ?ow can be solved and non elementary integrals evaluated. The in?nite process of summing of an in?nite series is a puzzle for centuries convergence and divergence of in?nite series plays an important role in engineering applications. 1 2 1 4 1 8 1 16 2.1 SEQUENCES A sequence is a function from the domain set of natural numbersN to any setS. Real sequence isafunctionfromN to R, the set of real numbers; denoted by f :N ? R. Thus the real sequence f is set of all ordered pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs (n,f (n)) with n a positive integer. Notation: Sincethedomainofasequenceisalways thesame(thesetofpositiveintegers)asequencemay be written as{f (n)} instead of{n,f (n)}. Examples: 1. n, 1 n = 1 n = 1, 1 2 , 1 3 , 1 4 ,... 1 n ... 2. n, 1 2 n-1 = 1 2 n-1 = 1, 1 2 , 1 2 2 , 1 2 3 , 1 2 4 ,..., 1 2 n-1 ,... constant sequence where range is singleton set{c}, c= constant. Example: {3, 3, 3, 3,...} Null sequence{0, 0, 0,..., 0,...} A sequence is also denoted by{a n } whose ordinate y = a n at the abscissax = n. Thus in a sequence for each positive integern, a numbera n is assigned and is denoted as a n or( a n )or {a n }={a(1),a(2),a(3),...,a(n),...} ={a 1 ,a 2 ,a 3 ,...,a n ,...} Herea 1 ,a 2 ,a 3 ,...a n , are known as the ?rst, second, third and nth terms of the sequence. In?nite sequence is a sequence in which the numberoftermsisin?nite,andisdenotedby {a n } 8 n=1 . Ontheotherhand,?nitesequencedenotedby{a n } m n=1 containsonlya?nitenumberofterms( m=?nite). Bounded sequence A sequence {a n } is said to be bounded if there exists numbers m and M such thatm<a n <M forevery n, otherwise it is said to be unbounded. Monotonic sequence A sequence{a n }issaidtobe a. monotonically increasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· 2.1 chap-02 B.V.Ramana August 30, 2006 10:15 2.2 ENGINEERING MATHEMATICS b. monotonically decreasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· c.monotonic if it is eithermonotonically increasing or monotonically decreasing. Example: 1 n = 1, 1 2 , 1 3 , 1 4 , 1 5 ,... bounded since 0<a n = 1 n < 1 and monotonically decreasing. Example: {2 n }={2, 2 2 , 2 3 , 2 4 ,...} unbounded since 2 n becomes larger and larger asn comes large and monotonically increasing. 2.2 LIMIT OF A SEQUENCE Considerasequence {a n }= 3+ 1 n . Plotting the values n: 1 2 4 5 10 50 100 1000 10000 100000... a n : 4 3.5 3.25 3.2 3.1 3.02 3.01 3.001 3.0001 3.00001... As n increases, a n = 3+ 1 n becomescloserto3. Thus the difference (or distance) between 3+ 1 n and 3becomessmallerandsmalleras n becomes larger and larger i.e., we can make 3+ 1 3 and3asclose as we please, by choosing an appropriately (suf?- ciently) large value forn, i.e., the terms of a sequence clusteraroundthis(limit)point.Howevernotethat 3+ 1 n =3foranyvalueof n. Limit: A number L is said to be a limit of a sequence{a n } and is denoted as lim As n?8 a n = lim n?8 a n = lim a n = L ifforevery > 0 there existsN such that |a n -L|< forall n=N. Note: Asequencemayhaveauniquelimitormay havemorethanonelimitormaynothavealimitatall. Result: A monotonic sequence always has a limit (maybe?niteorin?nite). 2.3 CONVERGENCE, DIVERGENCE AND OSCILLATION OF A SEQUENCE Convergent A sequence{a n } is said to be conver- gent if it has a ?nite limit i.e., lim n?8 a n = L= ?nite unique limit value. Divergent If lim n?8 a n = in?nite=±8. Oscillatory If limit ofa n is not unique (oscillates ?nitely) or±8 (oscillates in?nitely). Examples: 1. 1 n 2 convergent since lim n?8 1 n 2 = 0=?nite unique 2. {n:} divergent since lim n?8 n :=8 3. {(-1) n } oscillates ?nitely, since lim n?8 (-1) n = 1,n even -1,n odd. 4. {(-1) n ·n 2 } oscillates in?nitely, since limit=±8. Result1: If sequence{a n } converges to limitL and {b n } converges toL * then a. {a n +b n } converges toL+L * b. {ca n } converges toCL c. {a n ·b n } converges toL·L * d. { a n b n } converges to L L * , providedL * = 0. Result2: Every convergent sequence is bounded. Example: 1 n is convergent and is bounded a n = 1 n < 1,forevery n. Result3: The converse is not true i.e., a bounded sequence may not be convergent. Example:{(-1) n } is oscillatory (has more than one limit but is bounded since-1= (-1) n = 1. Result4: A bounded monotonic sequence is con- vergent. Example: 1 n 2 is bounded since 1 n 2 =1forevery n and monotonically decreasing since 1 n 2 > 1 (n+1) 2 for every n. Hence the sequence is convergent because lim n?8 a n = lim n?8 1 n 2 = 0= ?nite. Useful Standard Limits 1. a. lim n?8 1 n = 0, b. lim n?8 1 n 2 = 0, c. lim n?8 1 v n = 0 2. lim n?8 n 1/n = 1 3. lim n?8 logn n = 0 4. lim n?8 1+ x n n = e x , forany x 5. lim n?8 x 1/n =1forx> 0 6. (a) lim n?8 x n =0for |x| <1i.e. - 1<x<1. (b) lim n?8 x n n! =0 forany x. In formulas (5) and 6(b)x remains ?xed asn?8 WORKED OUT EXAMPLES Determine the nature of the following sequences whose nth term a n is Page 3 chap-02 B.V.Ramana August 30, 2006 10:15 Chapter2 Sequences and Series INTRODUCTION The study of convergence and divergence of a se- quence, which is an ordered list of things, is a prereq- uisit for in?nite series. The unit square in the ?gure can be expressed as an in?nite (geometric) series 1= 1 2 + 1 4 + 1 8 + 1 16 + 1 32 +··· Several functions can be expressed as “in?nite polynomials” (known as “powerser ies”) using the concept of in?nite series. By Fourier series, certain functions can be represented as an in?nite sum of trigonometric functions. Using in?nite series, differential equations in problems of signal transmission, chemical diffusion, vibration and heat ?ow can be solved and non elementary integrals evaluated. The in?nite process of summing of an in?nite series is a puzzle for centuries convergence and divergence of in?nite series plays an important role in engineering applications. 1 2 1 4 1 8 1 16 2.1 SEQUENCES A sequence is a function from the domain set of natural numbersN to any setS. Real sequence isafunctionfromN to R, the set of real numbers; denoted by f :N ? R. Thus the real sequence f is set of all ordered pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs (n,f (n)) with n a positive integer. Notation: Sincethedomainofasequenceisalways thesame(thesetofpositiveintegers)asequencemay be written as{f (n)} instead of{n,f (n)}. Examples: 1. n, 1 n = 1 n = 1, 1 2 , 1 3 , 1 4 ,... 1 n ... 2. n, 1 2 n-1 = 1 2 n-1 = 1, 1 2 , 1 2 2 , 1 2 3 , 1 2 4 ,..., 1 2 n-1 ,... constant sequence where range is singleton set{c}, c= constant. Example: {3, 3, 3, 3,...} Null sequence{0, 0, 0,..., 0,...} A sequence is also denoted by{a n } whose ordinate y = a n at the abscissax = n. Thus in a sequence for each positive integern, a numbera n is assigned and is denoted as a n or( a n )or {a n }={a(1),a(2),a(3),...,a(n),...} ={a 1 ,a 2 ,a 3 ,...,a n ,...} Herea 1 ,a 2 ,a 3 ,...a n , are known as the ?rst, second, third and nth terms of the sequence. In?nite sequence is a sequence in which the numberoftermsisin?nite,andisdenotedby {a n } 8 n=1 . Ontheotherhand,?nitesequencedenotedby{a n } m n=1 containsonlya?nitenumberofterms( m=?nite). Bounded sequence A sequence {a n } is said to be bounded if there exists numbers m and M such thatm<a n <M forevery n, otherwise it is said to be unbounded. Monotonic sequence A sequence{a n }issaidtobe a. monotonically increasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· 2.1 chap-02 B.V.Ramana August 30, 2006 10:15 2.2 ENGINEERING MATHEMATICS b. monotonically decreasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· c.monotonic if it is eithermonotonically increasing or monotonically decreasing. Example: 1 n = 1, 1 2 , 1 3 , 1 4 , 1 5 ,... bounded since 0<a n = 1 n < 1 and monotonically decreasing. Example: {2 n }={2, 2 2 , 2 3 , 2 4 ,...} unbounded since 2 n becomes larger and larger asn comes large and monotonically increasing. 2.2 LIMIT OF A SEQUENCE Considerasequence {a n }= 3+ 1 n . Plotting the values n: 1 2 4 5 10 50 100 1000 10000 100000... a n : 4 3.5 3.25 3.2 3.1 3.02 3.01 3.001 3.0001 3.00001... As n increases, a n = 3+ 1 n becomescloserto3. Thus the difference (or distance) between 3+ 1 n and 3becomessmallerandsmalleras n becomes larger and larger i.e., we can make 3+ 1 3 and3asclose as we please, by choosing an appropriately (suf?- ciently) large value forn, i.e., the terms of a sequence clusteraroundthis(limit)point.Howevernotethat 3+ 1 n =3foranyvalueof n. Limit: A number L is said to be a limit of a sequence{a n } and is denoted as lim As n?8 a n = lim n?8 a n = lim a n = L ifforevery > 0 there existsN such that |a n -L|< forall n=N. Note: Asequencemayhaveauniquelimitormay havemorethanonelimitormaynothavealimitatall. Result: A monotonic sequence always has a limit (maybe?niteorin?nite). 2.3 CONVERGENCE, DIVERGENCE AND OSCILLATION OF A SEQUENCE Convergent A sequence{a n } is said to be conver- gent if it has a ?nite limit i.e., lim n?8 a n = L= ?nite unique limit value. Divergent If lim n?8 a n = in?nite=±8. Oscillatory If limit ofa n is not unique (oscillates ?nitely) or±8 (oscillates in?nitely). Examples: 1. 1 n 2 convergent since lim n?8 1 n 2 = 0=?nite unique 2. {n:} divergent since lim n?8 n :=8 3. {(-1) n } oscillates ?nitely, since lim n?8 (-1) n = 1,n even -1,n odd. 4. {(-1) n ·n 2 } oscillates in?nitely, since limit=±8. Result1: If sequence{a n } converges to limitL and {b n } converges toL * then a. {a n +b n } converges toL+L * b. {ca n } converges toCL c. {a n ·b n } converges toL·L * d. { a n b n } converges to L L * , providedL * = 0. Result2: Every convergent sequence is bounded. Example: 1 n is convergent and is bounded a n = 1 n < 1,forevery n. Result3: The converse is not true i.e., a bounded sequence may not be convergent. Example:{(-1) n } is oscillatory (has more than one limit but is bounded since-1= (-1) n = 1. Result4: A bounded monotonic sequence is con- vergent. Example: 1 n 2 is bounded since 1 n 2 =1forevery n and monotonically decreasing since 1 n 2 > 1 (n+1) 2 for every n. Hence the sequence is convergent because lim n?8 a n = lim n?8 1 n 2 = 0= ?nite. Useful Standard Limits 1. a. lim n?8 1 n = 0, b. lim n?8 1 n 2 = 0, c. lim n?8 1 v n = 0 2. lim n?8 n 1/n = 1 3. lim n?8 logn n = 0 4. lim n?8 1+ x n n = e x , forany x 5. lim n?8 x 1/n =1forx> 0 6. (a) lim n?8 x n =0for |x| <1i.e. - 1<x<1. (b) lim n?8 x n n! =0 forany x. In formulas (5) and 6(b)x remains ?xed asn?8 WORKED OUT EXAMPLES Determine the nature of the following sequences whose nth term a n is chap-02 B.V.Ramana August 30, 2006 10:15 SEQUENCESANDSERIES 2.3 Example1: a n = n 2 -n 2n 2 +n Solution: lim n?8 a n = lim n?8 n 2 -n 2n 2 +n = lim n?8 1- 1 n 2+ 1 n = 1 2 sequence is convergent since the limit of the se- quence is unique and ?nite. Example2: a n = tanh n. Solution: lim n?8 a n = lim n?8 tanh n= lim n?8 sinh n cosh n = lim n?8 e n -e -n e n +e -n = lim n?8 e 2n - 1 e 2n + 1 = lim n?8 1- 1 e 2n 1+ 1 e 2n = 1 so convergent. Example3: a n = e n . Solution: lim n?8 e n =8 so divergent. Example4: a n = 2+ (-1) n . Solution: lim n?8 a 2n = lim n?8 {2+ (-1) 2n }= 2+ 1= 3 lim n?8 a 2n-1 = lim n?8 {2+ (-1) 2n-1 }= 2- 1= 1 sequence oscillates ?nitely since it has more than one ?nite (two) limits. EXERCISE 1. 2n+1 1-3n Ans. convergent, limit= -2 3 2. 1+ (-1) n n Ans. convergent, limit= 1 3. 1+(-1) n n Ans. convergent, limit= 0 4. sinn Ans. divergent, limit=8 5. lnn n Ans. convergent, limit= 0 Hint: Apply L’ Hospital’s rule. 6. 1 3 n Ans. convergent, limit= 3 2 7. (-1) n-1 n 3 n Ans. convergent 8. n n+1 2 Ans. convergent 9. (n+1) 2 (n+1)! Ans. convergent 10. 2n Ans. divergent, limit=8 11. 1+ 1 n Ans. convergent, limit= 1 12. [n+ (-1) n ] -1 Ans. convergent. 2.4 INFINITE SERIES Differential Equations are frequently solved by using in?nite series. Fourier series, Fourier-Bessel series, etc. expansions involve in?nite series. Tran- scendental functions (trigonometric, exponential, logarithmic, hyperpolic, etc.) can be expressed conveniently in terms of in?nite series. Many prob- lems that cannot be solved in terms of elementary (algebraic and transcendental) functions can also be solved in terms of in?nite series. Series Given a sequence of numbers u 1 ,u 2 ,u 3 ,...u n ,... the expression u 1 +u 2 +u 3 +···+u n +··· (1) which is the sum of the terms of the sequence, is known as a numerical series or simply “series”. The numbers u 1 ,u 2 ,u 3 ,...u n are known as the ?rst, second, third,...,nth term of the series (1). In?nite Series Ifthenumberoftermsintheseries(1)isin?nite,then the series is called an in?nite series (otherwise ?nite series when the number of terms is ?nite). In?nite series (1) is usually denoted as 8 n=1 u n or u n (1) Themainaimofthischapteristostudythenature(or behaviour) of convergence, divergence or oscillation ofagivenin?niteseries.Forthispurpose,de?ne{S n } Page 4 chap-02 B.V.Ramana August 30, 2006 10:15 Chapter2 Sequences and Series INTRODUCTION The study of convergence and divergence of a se- quence, which is an ordered list of things, is a prereq- uisit for in?nite series. The unit square in the ?gure can be expressed as an in?nite (geometric) series 1= 1 2 + 1 4 + 1 8 + 1 16 + 1 32 +··· Several functions can be expressed as “in?nite polynomials” (known as “powerser ies”) using the concept of in?nite series. By Fourier series, certain functions can be represented as an in?nite sum of trigonometric functions. Using in?nite series, differential equations in problems of signal transmission, chemical diffusion, vibration and heat ?ow can be solved and non elementary integrals evaluated. The in?nite process of summing of an in?nite series is a puzzle for centuries convergence and divergence of in?nite series plays an important role in engineering applications. 1 2 1 4 1 8 1 16 2.1 SEQUENCES A sequence is a function from the domain set of natural numbersN to any setS. Real sequence isafunctionfromN to R, the set of real numbers; denoted by f :N ? R. Thus the real sequence f is set of all ordered pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs (n,f (n)) with n a positive integer. Notation: Sincethedomainofasequenceisalways thesame(thesetofpositiveintegers)asequencemay be written as{f (n)} instead of{n,f (n)}. Examples: 1. n, 1 n = 1 n = 1, 1 2 , 1 3 , 1 4 ,... 1 n ... 2. n, 1 2 n-1 = 1 2 n-1 = 1, 1 2 , 1 2 2 , 1 2 3 , 1 2 4 ,..., 1 2 n-1 ,... constant sequence where range is singleton set{c}, c= constant. Example: {3, 3, 3, 3,...} Null sequence{0, 0, 0,..., 0,...} A sequence is also denoted by{a n } whose ordinate y = a n at the abscissax = n. Thus in a sequence for each positive integern, a numbera n is assigned and is denoted as a n or( a n )or {a n }={a(1),a(2),a(3),...,a(n),...} ={a 1 ,a 2 ,a 3 ,...,a n ,...} Herea 1 ,a 2 ,a 3 ,...a n , are known as the ?rst, second, third and nth terms of the sequence. In?nite sequence is a sequence in which the numberoftermsisin?nite,andisdenotedby {a n } 8 n=1 . Ontheotherhand,?nitesequencedenotedby{a n } m n=1 containsonlya?nitenumberofterms( m=?nite). Bounded sequence A sequence {a n } is said to be bounded if there exists numbers m and M such thatm<a n <M forevery n, otherwise it is said to be unbounded. Monotonic sequence A sequence{a n }issaidtobe a. monotonically increasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· 2.1 chap-02 B.V.Ramana August 30, 2006 10:15 2.2 ENGINEERING MATHEMATICS b. monotonically decreasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· c.monotonic if it is eithermonotonically increasing or monotonically decreasing. Example: 1 n = 1, 1 2 , 1 3 , 1 4 , 1 5 ,... bounded since 0<a n = 1 n < 1 and monotonically decreasing. Example: {2 n }={2, 2 2 , 2 3 , 2 4 ,...} unbounded since 2 n becomes larger and larger asn comes large and monotonically increasing. 2.2 LIMIT OF A SEQUENCE Considerasequence {a n }= 3+ 1 n . Plotting the values n: 1 2 4 5 10 50 100 1000 10000 100000... a n : 4 3.5 3.25 3.2 3.1 3.02 3.01 3.001 3.0001 3.00001... As n increases, a n = 3+ 1 n becomescloserto3. Thus the difference (or distance) between 3+ 1 n and 3becomessmallerandsmalleras n becomes larger and larger i.e., we can make 3+ 1 3 and3asclose as we please, by choosing an appropriately (suf?- ciently) large value forn, i.e., the terms of a sequence clusteraroundthis(limit)point.Howevernotethat 3+ 1 n =3foranyvalueof n. Limit: A number L is said to be a limit of a sequence{a n } and is denoted as lim As n?8 a n = lim n?8 a n = lim a n = L ifforevery > 0 there existsN such that |a n -L|< forall n=N. Note: Asequencemayhaveauniquelimitormay havemorethanonelimitormaynothavealimitatall. Result: A monotonic sequence always has a limit (maybe?niteorin?nite). 2.3 CONVERGENCE, DIVERGENCE AND OSCILLATION OF A SEQUENCE Convergent A sequence{a n } is said to be conver- gent if it has a ?nite limit i.e., lim n?8 a n = L= ?nite unique limit value. Divergent If lim n?8 a n = in?nite=±8. Oscillatory If limit ofa n is not unique (oscillates ?nitely) or±8 (oscillates in?nitely). Examples: 1. 1 n 2 convergent since lim n?8 1 n 2 = 0=?nite unique 2. {n:} divergent since lim n?8 n :=8 3. {(-1) n } oscillates ?nitely, since lim n?8 (-1) n = 1,n even -1,n odd. 4. {(-1) n ·n 2 } oscillates in?nitely, since limit=±8. Result1: If sequence{a n } converges to limitL and {b n } converges toL * then a. {a n +b n } converges toL+L * b. {ca n } converges toCL c. {a n ·b n } converges toL·L * d. { a n b n } converges to L L * , providedL * = 0. Result2: Every convergent sequence is bounded. Example: 1 n is convergent and is bounded a n = 1 n < 1,forevery n. Result3: The converse is not true i.e., a bounded sequence may not be convergent. Example:{(-1) n } is oscillatory (has more than one limit but is bounded since-1= (-1) n = 1. Result4: A bounded monotonic sequence is con- vergent. Example: 1 n 2 is bounded since 1 n 2 =1forevery n and monotonically decreasing since 1 n 2 > 1 (n+1) 2 for every n. Hence the sequence is convergent because lim n?8 a n = lim n?8 1 n 2 = 0= ?nite. Useful Standard Limits 1. a. lim n?8 1 n = 0, b. lim n?8 1 n 2 = 0, c. lim n?8 1 v n = 0 2. lim n?8 n 1/n = 1 3. lim n?8 logn n = 0 4. lim n?8 1+ x n n = e x , forany x 5. lim n?8 x 1/n =1forx> 0 6. (a) lim n?8 x n =0for |x| <1i.e. - 1<x<1. (b) lim n?8 x n n! =0 forany x. In formulas (5) and 6(b)x remains ?xed asn?8 WORKED OUT EXAMPLES Determine the nature of the following sequences whose nth term a n is chap-02 B.V.Ramana August 30, 2006 10:15 SEQUENCESANDSERIES 2.3 Example1: a n = n 2 -n 2n 2 +n Solution: lim n?8 a n = lim n?8 n 2 -n 2n 2 +n = lim n?8 1- 1 n 2+ 1 n = 1 2 sequence is convergent since the limit of the se- quence is unique and ?nite. Example2: a n = tanh n. Solution: lim n?8 a n = lim n?8 tanh n= lim n?8 sinh n cosh n = lim n?8 e n -e -n e n +e -n = lim n?8 e 2n - 1 e 2n + 1 = lim n?8 1- 1 e 2n 1+ 1 e 2n = 1 so convergent. Example3: a n = e n . Solution: lim n?8 e n =8 so divergent. Example4: a n = 2+ (-1) n . Solution: lim n?8 a 2n = lim n?8 {2+ (-1) 2n }= 2+ 1= 3 lim n?8 a 2n-1 = lim n?8 {2+ (-1) 2n-1 }= 2- 1= 1 sequence oscillates ?nitely since it has more than one ?nite (two) limits. EXERCISE 1. 2n+1 1-3n Ans. convergent, limit= -2 3 2. 1+ (-1) n n Ans. convergent, limit= 1 3. 1+(-1) n n Ans. convergent, limit= 0 4. sinn Ans. divergent, limit=8 5. lnn n Ans. convergent, limit= 0 Hint: Apply L’ Hospital’s rule. 6. 1 3 n Ans. convergent, limit= 3 2 7. (-1) n-1 n 3 n Ans. convergent 8. n n+1 2 Ans. convergent 9. (n+1) 2 (n+1)! Ans. convergent 10. 2n Ans. divergent, limit=8 11. 1+ 1 n Ans. convergent, limit= 1 12. [n+ (-1) n ] -1 Ans. convergent. 2.4 INFINITE SERIES Differential Equations are frequently solved by using in?nite series. Fourier series, Fourier-Bessel series, etc. expansions involve in?nite series. Tran- scendental functions (trigonometric, exponential, logarithmic, hyperpolic, etc.) can be expressed conveniently in terms of in?nite series. Many prob- lems that cannot be solved in terms of elementary (algebraic and transcendental) functions can also be solved in terms of in?nite series. Series Given a sequence of numbers u 1 ,u 2 ,u 3 ,...u n ,... the expression u 1 +u 2 +u 3 +···+u n +··· (1) which is the sum of the terms of the sequence, is known as a numerical series or simply “series”. The numbers u 1 ,u 2 ,u 3 ,...u n are known as the ?rst, second, third,...,nth term of the series (1). In?nite Series Ifthenumberoftermsintheseries(1)isin?nite,then the series is called an in?nite series (otherwise ?nite series when the number of terms is ?nite). In?nite series (1) is usually denoted as 8 n=1 u n or u n (1) Themainaimofthischapteristostudythenature(or behaviour) of convergence, divergence or oscillation ofagivenin?niteseries.Forthispurpose,de?ne{S n } chap-02 B.V.Ramana August 30, 2006 10:15 2.4 ENGINEERING MATHEMATICS the sequence of partial sums as S 1 =u 1 S 2 =u 1 +u 2 S 3 =u 1 +u 2 +u 3 . . . S n =u 1 +u 2 +u 3 +···+u n = n k=1 u k Here S n is known as the nth partial sum of the series, i.e., it is the sum of the ?rst n terms of the series (1). Convergence An in?nite series 8 n=1 u n is said to be convergent if 8 n=1 u n = lim n?8 n k=1 u k = lim n?8 S n =?nite limit value=S HereS is known as the sum (value) of the series (1). Divergence If lim n?8 S n does not exist (i.e., lim n?8 S n =±8) then series (1) is said to be divergent. Oscillation When lim n?8 S n tends to more than one limit (non unique) orto ±8 then series (1) is said to be os- cillatory. Thus the behaviour of convergence, diver- genceoroscillationofaseriesisthebahaviourofits sequence of partial sums{S n }. Example: 1+ 1 4 + 1 16 + 1 64 +··· Hereu n = 1 4 n-1 so lim n?8 S n = lim n?8 1- 1 4 n 1- 1 4 = lim n?8 4 3 1- 1 4 n = 4 3 = ?nite, series converges. Example: 1 2 + 2 2 + 3 2 +···+n 2 +··· lim n?8 S n = lim n?8 n(n+1)(2n+1) 6 =8, series diverges. Example: 7- 4- 3+ 7- 4- 3+ 7- 4- 3+··· lim n?8 S n =0 or7 or3 accor ding as the numberof terms is 3m, 3m+1or3m+ 2. Since the limit is not unique, series oscillates (?nitely). Example: 1-2+3-4+···+(-1) n-1 n+··· lim n?8 S n =- n 2 =-8 if n is even lim n?8 S n = n+ 1 2 =+8 if n is odd series oscillates (in?nitely). Some General Properties of Series 1. If a series u n converges to a sum s then the seriesc u n also converges to the sumcs, where c is a constant. 2. Iftheseries u n and v n converges to the sums s ands respectively then the series (u n +v n ) and (u n -v n ) also converge to s +s and s –s respectively. Addition or subtraction of two series is done by termwise addition or termwise subtraction. 3. The convergence of a series is not affected by the suppression (deletion) or addition of a ?nite numberofitsterms,sincethedeletionoraddition ofthesumofthese?nitenumberofterms(which isa?nitequantity)doesnotalterthebehaviour of the sum of the series. 2.5 NECESSARY CONDITION FOR CONVERGENCE Necessary condition for convergence of a series u n is that, its nth term u n approaches zero as n becomes in?nite i.e., If series converges, then lim n?8 u n = 0. Important Note: Thisisnotatestforconvergence. Proof: Let s be the sum of this convergent series. Also letS n andS n-1 be the nth and (n- 1)th partial sums of the given series so that u n = S n -S n-1 Taking limit, we have lim n?8 u n = lim n?8 (S n -S n-1 )= lim n?8 S n - lim n?8 S n-1 =s-s = 0. Note 1: The converse of the above result is not true, i.e., the above result is not a suf?cient condition. From the fact that the nth term u n approaches zero, Page 5 chap-02 B.V.Ramana August 30, 2006 10:15 Chapter2 Sequences and Series INTRODUCTION The study of convergence and divergence of a se- quence, which is an ordered list of things, is a prereq- uisit for in?nite series. The unit square in the ?gure can be expressed as an in?nite (geometric) series 1= 1 2 + 1 4 + 1 8 + 1 16 + 1 32 +··· Several functions can be expressed as “in?nite polynomials” (known as “powerser ies”) using the concept of in?nite series. By Fourier series, certain functions can be represented as an in?nite sum of trigonometric functions. Using in?nite series, differential equations in problems of signal transmission, chemical diffusion, vibration and heat ?ow can be solved and non elementary integrals evaluated. The in?nite process of summing of an in?nite series is a puzzle for centuries convergence and divergence of in?nite series plays an important role in engineering applications. 1 2 1 4 1 8 1 16 2.1 SEQUENCES A sequence is a function from the domain set of natural numbersN to any setS. Real sequence isafunctionfromN to R, the set of real numbers; denoted by f :N ? R. Thus the real sequence f is set of all ordered pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs (n,f (n)) with n a positive integer. Notation: Sincethedomainofasequenceisalways thesame(thesetofpositiveintegers)asequencemay be written as{f (n)} instead of{n,f (n)}. Examples: 1. n, 1 n = 1 n = 1, 1 2 , 1 3 , 1 4 ,... 1 n ... 2. n, 1 2 n-1 = 1 2 n-1 = 1, 1 2 , 1 2 2 , 1 2 3 , 1 2 4 ,..., 1 2 n-1 ,... constant sequence where range is singleton set{c}, c= constant. Example: {3, 3, 3, 3,...} Null sequence{0, 0, 0,..., 0,...} A sequence is also denoted by{a n } whose ordinate y = a n at the abscissax = n. Thus in a sequence for each positive integern, a numbera n is assigned and is denoted as a n or( a n )or {a n }={a(1),a(2),a(3),...,a(n),...} ={a 1 ,a 2 ,a 3 ,...,a n ,...} Herea 1 ,a 2 ,a 3 ,...a n , are known as the ?rst, second, third and nth terms of the sequence. In?nite sequence is a sequence in which the numberoftermsisin?nite,andisdenotedby {a n } 8 n=1 . Ontheotherhand,?nitesequencedenotedby{a n } m n=1 containsonlya?nitenumberofterms( m=?nite). Bounded sequence A sequence {a n } is said to be bounded if there exists numbers m and M such thatm<a n <M forevery n, otherwise it is said to be unbounded. Monotonic sequence A sequence{a n }issaidtobe a. monotonically increasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· 2.1 chap-02 B.V.Ramana August 30, 2006 10:15 2.2 ENGINEERING MATHEMATICS b. monotonically decreasing if a n+1 = a n for every n i.e., a 1 = a 2 = a 3 =···= a n = a n+1 =··· c.monotonic if it is eithermonotonically increasing or monotonically decreasing. Example: 1 n = 1, 1 2 , 1 3 , 1 4 , 1 5 ,... bounded since 0<a n = 1 n < 1 and monotonically decreasing. Example: {2 n }={2, 2 2 , 2 3 , 2 4 ,...} unbounded since 2 n becomes larger and larger asn comes large and monotonically increasing. 2.2 LIMIT OF A SEQUENCE Considerasequence {a n }= 3+ 1 n . Plotting the values n: 1 2 4 5 10 50 100 1000 10000 100000... a n : 4 3.5 3.25 3.2 3.1 3.02 3.01 3.001 3.0001 3.00001... As n increases, a n = 3+ 1 n becomescloserto3. Thus the difference (or distance) between 3+ 1 n and 3becomessmallerandsmalleras n becomes larger and larger i.e., we can make 3+ 1 3 and3asclose as we please, by choosing an appropriately (suf?- ciently) large value forn, i.e., the terms of a sequence clusteraroundthis(limit)point.Howevernotethat 3+ 1 n =3foranyvalueof n. Limit: A number L is said to be a limit of a sequence{a n } and is denoted as lim As n?8 a n = lim n?8 a n = lim a n = L ifforevery > 0 there existsN such that |a n -L|< forall n=N. Note: Asequencemayhaveauniquelimitormay havemorethanonelimitormaynothavealimitatall. Result: A monotonic sequence always has a limit (maybe?niteorin?nite). 2.3 CONVERGENCE, DIVERGENCE AND OSCILLATION OF A SEQUENCE Convergent A sequence{a n } is said to be conver- gent if it has a ?nite limit i.e., lim n?8 a n = L= ?nite unique limit value. Divergent If lim n?8 a n = in?nite=±8. Oscillatory If limit ofa n is not unique (oscillates ?nitely) or±8 (oscillates in?nitely). Examples: 1. 1 n 2 convergent since lim n?8 1 n 2 = 0=?nite unique 2. {n:} divergent since lim n?8 n :=8 3. {(-1) n } oscillates ?nitely, since lim n?8 (-1) n = 1,n even -1,n odd. 4. {(-1) n ·n 2 } oscillates in?nitely, since limit=±8. Result1: If sequence{a n } converges to limitL and {b n } converges toL * then a. {a n +b n } converges toL+L * b. {ca n } converges toCL c. {a n ·b n } converges toL·L * d. { a n b n } converges to L L * , providedL * = 0. Result2: Every convergent sequence is bounded. Example: 1 n is convergent and is bounded a n = 1 n < 1,forevery n. Result3: The converse is not true i.e., a bounded sequence may not be convergent. Example:{(-1) n } is oscillatory (has more than one limit but is bounded since-1= (-1) n = 1. Result4: A bounded monotonic sequence is con- vergent. Example: 1 n 2 is bounded since 1 n 2 =1forevery n and monotonically decreasing since 1 n 2 > 1 (n+1) 2 for every n. Hence the sequence is convergent because lim n?8 a n = lim n?8 1 n 2 = 0= ?nite. Useful Standard Limits 1. a. lim n?8 1 n = 0, b. lim n?8 1 n 2 = 0, c. lim n?8 1 v n = 0 2. lim n?8 n 1/n = 1 3. lim n?8 logn n = 0 4. lim n?8 1+ x n n = e x , forany x 5. lim n?8 x 1/n =1forx> 0 6. (a) lim n?8 x n =0for |x| <1i.e. - 1<x<1. (b) lim n?8 x n n! =0 forany x. In formulas (5) and 6(b)x remains ?xed asn?8 WORKED OUT EXAMPLES Determine the nature of the following sequences whose nth term a n is chap-02 B.V.Ramana August 30, 2006 10:15 SEQUENCESANDSERIES 2.3 Example1: a n = n 2 -n 2n 2 +n Solution: lim n?8 a n = lim n?8 n 2 -n 2n 2 +n = lim n?8 1- 1 n 2+ 1 n = 1 2 sequence is convergent since the limit of the se- quence is unique and ?nite. Example2: a n = tanh n. Solution: lim n?8 a n = lim n?8 tanh n= lim n?8 sinh n cosh n = lim n?8 e n -e -n e n +e -n = lim n?8 e 2n - 1 e 2n + 1 = lim n?8 1- 1 e 2n 1+ 1 e 2n = 1 so convergent. Example3: a n = e n . Solution: lim n?8 e n =8 so divergent. Example4: a n = 2+ (-1) n . Solution: lim n?8 a 2n = lim n?8 {2+ (-1) 2n }= 2+ 1= 3 lim n?8 a 2n-1 = lim n?8 {2+ (-1) 2n-1 }= 2- 1= 1 sequence oscillates ?nitely since it has more than one ?nite (two) limits. EXERCISE 1. 2n+1 1-3n Ans. convergent, limit= -2 3 2. 1+ (-1) n n Ans. convergent, limit= 1 3. 1+(-1) n n Ans. convergent, limit= 0 4. sinn Ans. divergent, limit=8 5. lnn n Ans. convergent, limit= 0 Hint: Apply L’ Hospital’s rule. 6. 1 3 n Ans. convergent, limit= 3 2 7. (-1) n-1 n 3 n Ans. convergent 8. n n+1 2 Ans. convergent 9. (n+1) 2 (n+1)! Ans. convergent 10. 2n Ans. divergent, limit=8 11. 1+ 1 n Ans. convergent, limit= 1 12. [n+ (-1) n ] -1 Ans. convergent. 2.4 INFINITE SERIES Differential Equations are frequently solved by using in?nite series. Fourier series, Fourier-Bessel series, etc. expansions involve in?nite series. Tran- scendental functions (trigonometric, exponential, logarithmic, hyperpolic, etc.) can be expressed conveniently in terms of in?nite series. Many prob- lems that cannot be solved in terms of elementary (algebraic and transcendental) functions can also be solved in terms of in?nite series. Series Given a sequence of numbers u 1 ,u 2 ,u 3 ,...u n ,... the expression u 1 +u 2 +u 3 +···+u n +··· (1) which is the sum of the terms of the sequence, is known as a numerical series or simply “series”. The numbers u 1 ,u 2 ,u 3 ,...u n are known as the ?rst, second, third,...,nth term of the series (1). In?nite Series Ifthenumberoftermsintheseries(1)isin?nite,then the series is called an in?nite series (otherwise ?nite series when the number of terms is ?nite). In?nite series (1) is usually denoted as 8 n=1 u n or u n (1) Themainaimofthischapteristostudythenature(or behaviour) of convergence, divergence or oscillation ofagivenin?niteseries.Forthispurpose,de?ne{S n } chap-02 B.V.Ramana August 30, 2006 10:15 2.4 ENGINEERING MATHEMATICS the sequence of partial sums as S 1 =u 1 S 2 =u 1 +u 2 S 3 =u 1 +u 2 +u 3 . . . S n =u 1 +u 2 +u 3 +···+u n = n k=1 u k Here S n is known as the nth partial sum of the series, i.e., it is the sum of the ?rst n terms of the series (1). Convergence An in?nite series 8 n=1 u n is said to be convergent if 8 n=1 u n = lim n?8 n k=1 u k = lim n?8 S n =?nite limit value=S HereS is known as the sum (value) of the series (1). Divergence If lim n?8 S n does not exist (i.e., lim n?8 S n =±8) then series (1) is said to be divergent. Oscillation When lim n?8 S n tends to more than one limit (non unique) orto ±8 then series (1) is said to be os- cillatory. Thus the behaviour of convergence, diver- genceoroscillationofaseriesisthebahaviourofits sequence of partial sums{S n }. Example: 1+ 1 4 + 1 16 + 1 64 +··· Hereu n = 1 4 n-1 so lim n?8 S n = lim n?8 1- 1 4 n 1- 1 4 = lim n?8 4 3 1- 1 4 n = 4 3 = ?nite, series converges. Example: 1 2 + 2 2 + 3 2 +···+n 2 +··· lim n?8 S n = lim n?8 n(n+1)(2n+1) 6 =8, series diverges. Example: 7- 4- 3+ 7- 4- 3+ 7- 4- 3+··· lim n?8 S n =0 or7 or3 accor ding as the numberof terms is 3m, 3m+1or3m+ 2. Since the limit is not unique, series oscillates (?nitely). Example: 1-2+3-4+···+(-1) n-1 n+··· lim n?8 S n =- n 2 =-8 if n is even lim n?8 S n = n+ 1 2 =+8 if n is odd series oscillates (in?nitely). Some General Properties of Series 1. If a series u n converges to a sum s then the seriesc u n also converges to the sumcs, where c is a constant. 2. Iftheseries u n and v n converges to the sums s ands respectively then the series (u n +v n ) and (u n -v n ) also converge to s +s and s –s respectively. Addition or subtraction of two series is done by termwise addition or termwise subtraction. 3. The convergence of a series is not affected by the suppression (deletion) or addition of a ?nite numberofitsterms,sincethedeletionoraddition ofthesumofthese?nitenumberofterms(which isa?nitequantity)doesnotalterthebehaviour of the sum of the series. 2.5 NECESSARY CONDITION FOR CONVERGENCE Necessary condition for convergence of a series u n is that, its nth term u n approaches zero as n becomes in?nite i.e., If series converges, then lim n?8 u n = 0. Important Note: Thisisnotatestforconvergence. Proof: Let s be the sum of this convergent series. Also letS n andS n-1 be the nth and (n- 1)th partial sums of the given series so that u n = S n -S n-1 Taking limit, we have lim n?8 u n = lim n?8 (S n -S n-1 )= lim n?8 S n - lim n?8 S n-1 =s-s = 0. Note 1: The converse of the above result is not true, i.e., the above result is not a suf?cient condition. From the fact that the nth term u n approaches zero, chap-02 B.V.Ramana August 30, 2006 10:15 SEQUENCESANDSERIES 2.5 it does not follow that the series converges, for the series may diverge. If lim n?8 u n = 0, then the series may converge or may diverge. Example: 1+ 1 2 + 1 3 + 1 4 +···+ 1 n +···isadi- vergent series although itsnth term approaches zero i.e., lim n?8 u n = lim n?8 1 n = 0 Note 2: Preliminary test for divergence. If thenth term of a series does not tend to zero as n?8, then the series diverges i.e., if lim n?8 u n = 0 then series diverges. Example: 1 2 + 2 3 + 3 4 + 4 5 +···+ n n+ 1 +··· Since lim n?8 u n = lim n?8 n n+1 = 1= 0 by the above preliminary test, the given series diverges. 2.6 STANDARD INFINITE SERIES: GEOMETRIC SERIES AND HARMONIC SERIES Geometric Series Test 8 n=0 ar n = a+ar+ar 2 +ar 3 +···+ar n-1 +···, with a =0(1) is a geometric series, whose terms form a geometric progression with the ?rst term a and the common ratio r.Forthisseries S n = a-ar n 1-r = a 1-r - ar n 1-r Case1: When|r| < 1 then lim n?8 r n = 0 so that lim n?8 S n = lim n?8 a 1-r - ar n 1-r = a 1-r - a 1-r · 0 = a 1-r = ?nite Hence geometric series (1) converges to the sum a 1-r when |r| < 1 i.e., in the interval -1 < r< 1. Case2: When|r| > 1 then lim n?8 r n =8 so that lim n?8 S n = lim n?8 a 1-r - ar n 1-r =±8 Thus series (1) diverges when|r| > 1 i.e., whenr>1orr<-1. Case3: If r = 1, the series (1) reduces to a+a+a+··· consequently lim n?8 S n = lim n?8 (na)=8 so series diverges. Case4: If r=-1, the series (1) reduces to a-a+a-a+··· In this case, Sn= 0, when n is even a, when n is odd Thus lim n?8 S n is not unique (more than one limit) hence the series diverges. Hence the geometric series converges only when |r| <1anddivergesforallothervaluesof r. Example: A ball is dropped from a height b feet from a ?at surface. Each time the ball hits the ground afterfalling a distance h it rebounds a distance rh where 0<r< 1 (Fig. 2.1). Fig. 2.1 Find the total distance the ball travels if b=4ft andr = 3 4 .Read More

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