F. Approximations
If x < 1, the terms of the above expansion go on decreasing and if x be very small, a stage may be reached when we may neglect the terms containing higher powers of x in the expansion. Thus, if x be so small that its squares and higher powers may be neglected then (1 + x)^{n} = 1 + nx, approximately,
This is an approximate value of (1 + x)^{n}
Ex.10 If x is so small such that its square and higher powers may be neglected then find the approximate value of
Sol.
Ex.11 The value of cube root of 1001 upto five decimal places is
Sol.
(1001)^{1/3} = (1000 + 1)^{1/3} =
= 10 {1 + 0.0003333 0.00000011 + ......} = 10.00333
Ex.12 The sum of is
Sol. Comparing with
1 + nx + + ...... ⇒ nx = 1/4 ...(i)
& or
⇒ ⇒
⇒ ...(ii) {by (i)}
putting the value of x in (i) ⇒ ⇒ n =
sum of series = (1 + x)^{n} = (1 1/2)^{1/2} = (1/2)^{1/2} =
G. Exponential series
(a) e is an irrational number lying between 2.7 & 2.8. Its value correct upto 10 places of decimal is 2.7182818284.
(b) Logarithms to the base 'e' are known as the Napierian system, so named after Napier, their inventor. They are also called Natural Logarithm.
(c)
where x may be any real or complex number &
(d)
(e)
H. Logarithmic series
(a) ln (1 + x) = x  where
(b) ln (1 x) = where
Remember :
(i)
(ii) e^{ln x} = x
(iii) ln2 = 0.693
(iv) ln 10 = 2.303
1. What are approximations and why are they important in mathematics? 
2. How are exponential series used in mathematics and reallife applications? 
3. Can approximations be used to solve equations or find roots of functions? 
4. What is the difference between linear and exponential approximations? 
5. Are there any limitations or drawbacks to using approximations in mathematics? 
157 videos210 docs132 tests

157 videos210 docs132 tests
