Series | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
3. Series 
3.1 Show that the series : (
?? ?? )
?? +(
?? ·?? ?? ·?? )
?? +?..+(
?? ·?? ·?? ….(?? ?? -?? )
?? ·?? ·?? ….?? ?? )
?? +? converges. 
(2009 : 15 Marks) 
Solution: 
(
1
3
)
2
+(
1·4
3·6
)
2
+?.+(
1.47….(3?? +2)
3.6.9….3?? )
2
?+?..
?? ?? ?=(
1.4.7….(3?? -2)
3.6.9…..3?? )
2
?? ?? +1
?=(
1.4.7….(3?? -2)(3?? +1)
3.6.9….(3?? )(3?? +3)
)
2
?? ?? ?? ?? +1
?=
(?? ?? +3)
2
.
(3?? +1)
2
?? ?? ?? ?? +1
-1?=
9?? 2
+9+18?? -9?? 2
-1-6?? (3?? +1)
2
?=
12?? +8
(3?? +1)
2
 
lim
?? ?8
??? (
?? ?? ?? ?? +1
-1)?= lim
?? ?8
?
12?? 2
+8?? (3?? +1)
2
?=
12
9
>1
 
By Rabee's test, series is convergent. 
3.2 Show that: ?????? ?? ??? ??
?? =?? 8
?
?? ?? ?? ?? ?? ?? +?? ?? =?
?? =?? 8
?
?? ?? ?? ?? +?? . Justify all steps of your answer by 
quoting the theorems you are using. 
(2009 : 15 Marks) 
Solution: 
To prove the required result, we use the following result : "The limit of the sum function 
of a series = the sum of the series of limits of functions", i.e., 
lim
?? ??? 0
???
8
?? =1
?? ?? (?? )=??
8
?? =1
lim
?? ??? 0
??? ?? (?? ) 
where ?
?? =1
8
??? ?? (?? ) converges uniformly in [?? ,?? ] and ?? 0
 is a point in [?? ,?? ]. 
Page 2


Edurev123 
3. Series 
3.1 Show that the series : (
?? ?? )
?? +(
?? ·?? ?? ·?? )
?? +?..+(
?? ·?? ·?? ….(?? ?? -?? )
?? ·?? ·?? ….?? ?? )
?? +? converges. 
(2009 : 15 Marks) 
Solution: 
(
1
3
)
2
+(
1·4
3·6
)
2
+?.+(
1.47….(3?? +2)
3.6.9….3?? )
2
?+?..
?? ?? ?=(
1.4.7….(3?? -2)
3.6.9…..3?? )
2
?? ?? +1
?=(
1.4.7….(3?? -2)(3?? +1)
3.6.9….(3?? )(3?? +3)
)
2
?? ?? ?? ?? +1
?=
(?? ?? +3)
2
.
(3?? +1)
2
?? ?? ?? ?? +1
-1?=
9?? 2
+9+18?? -9?? 2
-1-6?? (3?? +1)
2
?=
12?? +8
(3?? +1)
2
 
lim
?? ?8
??? (
?? ?? ?? ?? +1
-1)?= lim
?? ?8
?
12?? 2
+8?? (3?? +1)
2
?=
12
9
>1
 
By Rabee's test, series is convergent. 
3.2 Show that: ?????? ?? ??? ??
?? =?? 8
?
?? ?? ?? ?? ?? ?? +?? ?? =?
?? =?? 8
?
?? ?? ?? ?? +?? . Justify all steps of your answer by 
quoting the theorems you are using. 
(2009 : 15 Marks) 
Solution: 
To prove the required result, we use the following result : "The limit of the sum function 
of a series = the sum of the series of limits of functions", i.e., 
lim
?? ??? 0
???
8
?? =1
?? ?? (?? )=??
8
?? =1
lim
?? ??? 0
??? ?? (?? ) 
where ?
?? =1
8
??? ?? (?? ) converges uniformly in [?? ,?? ] and ?? 0
 is a point in [?? ,?? ]. 
So, as per the question, if we take ?? =0 and ?? =2 and ?? 0
=1 i.e., ?? 0
 is a point in [0,2], 
then 
lim
?? ?1
???
8
?? =1
?
?? 2
?? 2
?? 4
+?? 4
=??
8
?? =1
?
?? 2
?? 4
+?? 4
(??) 
But for this result, we need to prove that ?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformiv in [0,2] and 
where ?? 0
=1 is a point in [0,2]. 
Now to prove ?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformly in [0,2].?[?? 0
=1 is taken] 
Proof : This theorem/result can be proved by Drichlet's test. 
Let 
??
8
?? =1
??? ?? (?? )?=??
8
?? =1
??? ?? (?? )h
?? (?? )=??
8
?? =1
??? 2
?? 2
×
1
?? 4
+?? 4
?? ?? (?? )?=?? 2
?? 2
 
???????????????????????????????????????????????h
?? (?? )=
1
?? 4
+?? 4
???????????????????????????????????????????????????????????????????????????????(???? ) 
Drichlet's test states that 
(i) if there exists a real number ?? such that 
|?? ?? (?? )|=??
?? ?? =1
?? ?? (?? )<?? ??? ?[?? ,?? ],?? ??? and  
(ii) ?h
?? (?? )? is positive monotonic decreasing sequence converging uniformly to zero on 
[?? ,?? ], then the series ? ?? ?? (?? )h
?? (?? ) is uniformly convergent on [?? ,?? ]. 
From (ii), we know that 
|?? ?? (?? )|=??
?? ?? =1
?? ?? (?? )=??
?? ?? =1
?? 2
?? 2
=
?? (?? +1)(2?? +1)?? 2
6
<?? ??? ?[0,2],?? ??? 
(where ?? is a real number) 
Also, 
<h?? (?? )>=
1
?? 4
+?? 4
 
is a positive monotonic decreasing sequence converges uniformly to zero on [0,2]. 
Page 3


Edurev123 
3. Series 
3.1 Show that the series : (
?? ?? )
?? +(
?? ·?? ?? ·?? )
?? +?..+(
?? ·?? ·?? ….(?? ?? -?? )
?? ·?? ·?? ….?? ?? )
?? +? converges. 
(2009 : 15 Marks) 
Solution: 
(
1
3
)
2
+(
1·4
3·6
)
2
+?.+(
1.47….(3?? +2)
3.6.9….3?? )
2
?+?..
?? ?? ?=(
1.4.7….(3?? -2)
3.6.9…..3?? )
2
?? ?? +1
?=(
1.4.7….(3?? -2)(3?? +1)
3.6.9….(3?? )(3?? +3)
)
2
?? ?? ?? ?? +1
?=
(?? ?? +3)
2
.
(3?? +1)
2
?? ?? ?? ?? +1
-1?=
9?? 2
+9+18?? -9?? 2
-1-6?? (3?? +1)
2
?=
12?? +8
(3?? +1)
2
 
lim
?? ?8
??? (
?? ?? ?? ?? +1
-1)?= lim
?? ?8
?
12?? 2
+8?? (3?? +1)
2
?=
12
9
>1
 
By Rabee's test, series is convergent. 
3.2 Show that: ?????? ?? ??? ??
?? =?? 8
?
?? ?? ?? ?? ?? ?? +?? ?? =?
?? =?? 8
?
?? ?? ?? ?? +?? . Justify all steps of your answer by 
quoting the theorems you are using. 
(2009 : 15 Marks) 
Solution: 
To prove the required result, we use the following result : "The limit of the sum function 
of a series = the sum of the series of limits of functions", i.e., 
lim
?? ??? 0
???
8
?? =1
?? ?? (?? )=??
8
?? =1
lim
?? ??? 0
??? ?? (?? ) 
where ?
?? =1
8
??? ?? (?? ) converges uniformly in [?? ,?? ] and ?? 0
 is a point in [?? ,?? ]. 
So, as per the question, if we take ?? =0 and ?? =2 and ?? 0
=1 i.e., ?? 0
 is a point in [0,2], 
then 
lim
?? ?1
???
8
?? =1
?
?? 2
?? 2
?? 4
+?? 4
=??
8
?? =1
?
?? 2
?? 4
+?? 4
(??) 
But for this result, we need to prove that ?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformiv in [0,2] and 
where ?? 0
=1 is a point in [0,2]. 
Now to prove ?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformly in [0,2].?[?? 0
=1 is taken] 
Proof : This theorem/result can be proved by Drichlet's test. 
Let 
??
8
?? =1
??? ?? (?? )?=??
8
?? =1
??? ?? (?? )h
?? (?? )=??
8
?? =1
??? 2
?? 2
×
1
?? 4
+?? 4
?? ?? (?? )?=?? 2
?? 2
 
???????????????????????????????????????????????h
?? (?? )=
1
?? 4
+?? 4
???????????????????????????????????????????????????????????????????????????????(???? ) 
Drichlet's test states that 
(i) if there exists a real number ?? such that 
|?? ?? (?? )|=??
?? ?? =1
?? ?? (?? )<?? ??? ?[?? ,?? ],?? ??? and  
(ii) ?h
?? (?? )? is positive monotonic decreasing sequence converging uniformly to zero on 
[?? ,?? ], then the series ? ?? ?? (?? )h
?? (?? ) is uniformly convergent on [?? ,?? ]. 
From (ii), we know that 
|?? ?? (?? )|=??
?? ?? =1
?? ?? (?? )=??
?? ?? =1
?? 2
?? 2
=
?? (?? +1)(2?? +1)?? 2
6
<?? ??? ?[0,2],?? ??? 
(where ?? is a real number) 
Also, 
<h?? (?? )>=
1
?? 4
+?? 4
 
is a positive monotonic decreasing sequence converges uniformly to zero on [0,2]. 
??????
?? =1
8
??? ?? (?? )h
?? (?? )=?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformly in [0,2] 
From (i) and (iii), we get the desired result. 
3.3 Consider the series ?
?? =?? 8
?
?? ?? (?? +?? ?? )
?? . Find the values of ?? for which it is convergent 
and also the sum function. Is the convergence uniform? Justify your answer. 
(2010 : 15 Marks) 
Solution: 
Given series is ?
?? =0
8
?
?? 2
(1+?? 2
)
?? =?? ?? . 
Case 1 : If ?? =0 
then ???????????????????????????????????????????????????????????? ?? =0 
So it uniformly converges to zero. 
Case 2 : If ?? ?0 
then 
?? ?? =??
8
?? =0
?? 2
(1+?? 2
)
?? 
Now, 
?? ?? ?? ?? +1
=
?? 2
(1+?? 2
)
?? ?? 2
(1+?? 2
)
?? +1
=1+?? 2
>1???? ?0 
? It will converge as per D'Alembert test. 
??? ?? converges for all values of ?? . 
Now, 
?? ?? ?=??
8
?? =0
?
?? 2
(1+?? 2
)
?? =?? 2
??
8
?? =0
?
1
(1+?? 2
)
?? ?=?? 2
×
(1-(
1
1+?? 2
)
?? )
1-
1
1+?? 2
?=(1+?? 2
) as ?? ?8
 
Page 4


Edurev123 
3. Series 
3.1 Show that the series : (
?? ?? )
?? +(
?? ·?? ?? ·?? )
?? +?..+(
?? ·?? ·?? ….(?? ?? -?? )
?? ·?? ·?? ….?? ?? )
?? +? converges. 
(2009 : 15 Marks) 
Solution: 
(
1
3
)
2
+(
1·4
3·6
)
2
+?.+(
1.47….(3?? +2)
3.6.9….3?? )
2
?+?..
?? ?? ?=(
1.4.7….(3?? -2)
3.6.9…..3?? )
2
?? ?? +1
?=(
1.4.7….(3?? -2)(3?? +1)
3.6.9….(3?? )(3?? +3)
)
2
?? ?? ?? ?? +1
?=
(?? ?? +3)
2
.
(3?? +1)
2
?? ?? ?? ?? +1
-1?=
9?? 2
+9+18?? -9?? 2
-1-6?? (3?? +1)
2
?=
12?? +8
(3?? +1)
2
 
lim
?? ?8
??? (
?? ?? ?? ?? +1
-1)?= lim
?? ?8
?
12?? 2
+8?? (3?? +1)
2
?=
12
9
>1
 
By Rabee's test, series is convergent. 
3.2 Show that: ?????? ?? ??? ??
?? =?? 8
?
?? ?? ?? ?? ?? ?? +?? ?? =?
?? =?? 8
?
?? ?? ?? ?? +?? . Justify all steps of your answer by 
quoting the theorems you are using. 
(2009 : 15 Marks) 
Solution: 
To prove the required result, we use the following result : "The limit of the sum function 
of a series = the sum of the series of limits of functions", i.e., 
lim
?? ??? 0
???
8
?? =1
?? ?? (?? )=??
8
?? =1
lim
?? ??? 0
??? ?? (?? ) 
where ?
?? =1
8
??? ?? (?? ) converges uniformly in [?? ,?? ] and ?? 0
 is a point in [?? ,?? ]. 
So, as per the question, if we take ?? =0 and ?? =2 and ?? 0
=1 i.e., ?? 0
 is a point in [0,2], 
then 
lim
?? ?1
???
8
?? =1
?
?? 2
?? 2
?? 4
+?? 4
=??
8
?? =1
?
?? 2
?? 4
+?? 4
(??) 
But for this result, we need to prove that ?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformiv in [0,2] and 
where ?? 0
=1 is a point in [0,2]. 
Now to prove ?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformly in [0,2].?[?? 0
=1 is taken] 
Proof : This theorem/result can be proved by Drichlet's test. 
Let 
??
8
?? =1
??? ?? (?? )?=??
8
?? =1
??? ?? (?? )h
?? (?? )=??
8
?? =1
??? 2
?? 2
×
1
?? 4
+?? 4
?? ?? (?? )?=?? 2
?? 2
 
???????????????????????????????????????????????h
?? (?? )=
1
?? 4
+?? 4
???????????????????????????????????????????????????????????????????????????????(???? ) 
Drichlet's test states that 
(i) if there exists a real number ?? such that 
|?? ?? (?? )|=??
?? ?? =1
?? ?? (?? )<?? ??? ?[?? ,?? ],?? ??? and  
(ii) ?h
?? (?? )? is positive monotonic decreasing sequence converging uniformly to zero on 
[?? ,?? ], then the series ? ?? ?? (?? )h
?? (?? ) is uniformly convergent on [?? ,?? ]. 
From (ii), we know that 
|?? ?? (?? )|=??
?? ?? =1
?? ?? (?? )=??
?? ?? =1
?? 2
?? 2
=
?? (?? +1)(2?? +1)?? 2
6
<?? ??? ?[0,2],?? ??? 
(where ?? is a real number) 
Also, 
<h?? (?? )>=
1
?? 4
+?? 4
 
is a positive monotonic decreasing sequence converges uniformly to zero on [0,2]. 
??????
?? =1
8
??? ?? (?? )h
?? (?? )=?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformly in [0,2] 
From (i) and (iii), we get the desired result. 
3.3 Consider the series ?
?? =?? 8
?
?? ?? (?? +?? ?? )
?? . Find the values of ?? for which it is convergent 
and also the sum function. Is the convergence uniform? Justify your answer. 
(2010 : 15 Marks) 
Solution: 
Given series is ?
?? =0
8
?
?? 2
(1+?? 2
)
?? =?? ?? . 
Case 1 : If ?? =0 
then ???????????????????????????????????????????????????????????? ?? =0 
So it uniformly converges to zero. 
Case 2 : If ?? ?0 
then 
?? ?? =??
8
?? =0
?? 2
(1+?? 2
)
?? 
Now, 
?? ?? ?? ?? +1
=
?? 2
(1+?? 2
)
?? ?? 2
(1+?? 2
)
?? +1
=1+?? 2
>1???? ?0 
? It will converge as per D'Alembert test. 
??? ?? converges for all values of ?? . 
Now, 
?? ?? ?=??
8
?? =0
?
?? 2
(1+?? 2
)
?? =?? 2
??
8
?? =0
?
1
(1+?? 2
)
?? ?=?? 2
×
(1-(
1
1+?? 2
)
?? )
1-
1
1+?? 2
?=(1+?? 2
) as ?? ?8
 
??? ?? is finite. 
So, ?? ?? is uniformly convergent as |?? ?? |=?? ?? . 
3.4 Show that the series for which the sum of first ?? terms 
?? ?? (?? )=
????
?? +?? ?? ?? ?? ,?? =?? =?? 
cannot be differentiated term-by-term at ?? =?? . What happens at ?? ??? ? 
(2011: 15 marks) 
Solution: 
Given 
??
8
?? =1
?? ?? (?? )=?? ?? (?? )=
????
1+?? 2
?? 2
 
??lim
?? ?8
???
8
?? =1
??? ?? (?? )?= lim
?? ?8
??? ?? (?? )= lim
?? ?8
?
????
1+?? 2
?? 2
?= lim
?? ?8
?
?? 1
?? 2
+?? 2
=0,??? ?[0,1]
 
If we differentiate term-wise, we get 
??
?? ?? =1
??? ?? '
(?? )=?? ?? '
(?? )=
?? (1-?? 2
?? 2
)
(1+?? 2
?? 2
)
2
(??) 
At ?? =0 
??
?? ?? =1
?? ?? '
(?? )=?? ?? (0)=?? , which clearly does not tend to 0 as ?? ?8.  
????
?? ?? =1
?? ?? (?? )=?? ?? (?? )=
?? ?? 1+?? 2
?? 2
 can not be differentiated term by term at ?? =0.  
When ?? ?0, from (i) 
lim
?? ?8
???
?? ?? =1
?? ?? '
(?? )= lim
?? ?8
??? ?? '
(?? )=0= lim
?? ?8
???
?? ?? =1
?? ?? (?? ) 
Hence, ?
?? =1
8
??? ?? (?? ) can be differentiated term by term when ?? ?0. 
Page 5


Edurev123 
3. Series 
3.1 Show that the series : (
?? ?? )
?? +(
?? ·?? ?? ·?? )
?? +?..+(
?? ·?? ·?? ….(?? ?? -?? )
?? ·?? ·?? ….?? ?? )
?? +? converges. 
(2009 : 15 Marks) 
Solution: 
(
1
3
)
2
+(
1·4
3·6
)
2
+?.+(
1.47….(3?? +2)
3.6.9….3?? )
2
?+?..
?? ?? ?=(
1.4.7….(3?? -2)
3.6.9…..3?? )
2
?? ?? +1
?=(
1.4.7….(3?? -2)(3?? +1)
3.6.9….(3?? )(3?? +3)
)
2
?? ?? ?? ?? +1
?=
(?? ?? +3)
2
.
(3?? +1)
2
?? ?? ?? ?? +1
-1?=
9?? 2
+9+18?? -9?? 2
-1-6?? (3?? +1)
2
?=
12?? +8
(3?? +1)
2
 
lim
?? ?8
??? (
?? ?? ?? ?? +1
-1)?= lim
?? ?8
?
12?? 2
+8?? (3?? +1)
2
?=
12
9
>1
 
By Rabee's test, series is convergent. 
3.2 Show that: ?????? ?? ??? ??
?? =?? 8
?
?? ?? ?? ?? ?? ?? +?? ?? =?
?? =?? 8
?
?? ?? ?? ?? +?? . Justify all steps of your answer by 
quoting the theorems you are using. 
(2009 : 15 Marks) 
Solution: 
To prove the required result, we use the following result : "The limit of the sum function 
of a series = the sum of the series of limits of functions", i.e., 
lim
?? ??? 0
???
8
?? =1
?? ?? (?? )=??
8
?? =1
lim
?? ??? 0
??? ?? (?? ) 
where ?
?? =1
8
??? ?? (?? ) converges uniformly in [?? ,?? ] and ?? 0
 is a point in [?? ,?? ]. 
So, as per the question, if we take ?? =0 and ?? =2 and ?? 0
=1 i.e., ?? 0
 is a point in [0,2], 
then 
lim
?? ?1
???
8
?? =1
?
?? 2
?? 2
?? 4
+?? 4
=??
8
?? =1
?
?? 2
?? 4
+?? 4
(??) 
But for this result, we need to prove that ?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformiv in [0,2] and 
where ?? 0
=1 is a point in [0,2]. 
Now to prove ?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformly in [0,2].?[?? 0
=1 is taken] 
Proof : This theorem/result can be proved by Drichlet's test. 
Let 
??
8
?? =1
??? ?? (?? )?=??
8
?? =1
??? ?? (?? )h
?? (?? )=??
8
?? =1
??? 2
?? 2
×
1
?? 4
+?? 4
?? ?? (?? )?=?? 2
?? 2
 
???????????????????????????????????????????????h
?? (?? )=
1
?? 4
+?? 4
???????????????????????????????????????????????????????????????????????????????(???? ) 
Drichlet's test states that 
(i) if there exists a real number ?? such that 
|?? ?? (?? )|=??
?? ?? =1
?? ?? (?? )<?? ??? ?[?? ,?? ],?? ??? and  
(ii) ?h
?? (?? )? is positive monotonic decreasing sequence converging uniformly to zero on 
[?? ,?? ], then the series ? ?? ?? (?? )h
?? (?? ) is uniformly convergent on [?? ,?? ]. 
From (ii), we know that 
|?? ?? (?? )|=??
?? ?? =1
?? ?? (?? )=??
?? ?? =1
?? 2
?? 2
=
?? (?? +1)(2?? +1)?? 2
6
<?? ??? ?[0,2],?? ??? 
(where ?? is a real number) 
Also, 
<h?? (?? )>=
1
?? 4
+?? 4
 
is a positive monotonic decreasing sequence converges uniformly to zero on [0,2]. 
??????
?? =1
8
??? ?? (?? )h
?? (?? )=?
?? =1
8
?
?? 2
?? 2
?? 4
+?? 4
 converges uniformly in [0,2] 
From (i) and (iii), we get the desired result. 
3.3 Consider the series ?
?? =?? 8
?
?? ?? (?? +?? ?? )
?? . Find the values of ?? for which it is convergent 
and also the sum function. Is the convergence uniform? Justify your answer. 
(2010 : 15 Marks) 
Solution: 
Given series is ?
?? =0
8
?
?? 2
(1+?? 2
)
?? =?? ?? . 
Case 1 : If ?? =0 
then ???????????????????????????????????????????????????????????? ?? =0 
So it uniformly converges to zero. 
Case 2 : If ?? ?0 
then 
?? ?? =??
8
?? =0
?? 2
(1+?? 2
)
?? 
Now, 
?? ?? ?? ?? +1
=
?? 2
(1+?? 2
)
?? ?? 2
(1+?? 2
)
?? +1
=1+?? 2
>1???? ?0 
? It will converge as per D'Alembert test. 
??? ?? converges for all values of ?? . 
Now, 
?? ?? ?=??
8
?? =0
?
?? 2
(1+?? 2
)
?? =?? 2
??
8
?? =0
?
1
(1+?? 2
)
?? ?=?? 2
×
(1-(
1
1+?? 2
)
?? )
1-
1
1+?? 2
?=(1+?? 2
) as ?? ?8
 
??? ?? is finite. 
So, ?? ?? is uniformly convergent as |?? ?? |=?? ?? . 
3.4 Show that the series for which the sum of first ?? terms 
?? ?? (?? )=
????
?? +?? ?? ?? ?? ,?? =?? =?? 
cannot be differentiated term-by-term at ?? =?? . What happens at ?? ??? ? 
(2011: 15 marks) 
Solution: 
Given 
??
8
?? =1
?? ?? (?? )=?? ?? (?? )=
????
1+?? 2
?? 2
 
??lim
?? ?8
???
8
?? =1
??? ?? (?? )?= lim
?? ?8
??? ?? (?? )= lim
?? ?8
?
????
1+?? 2
?? 2
?= lim
?? ?8
?
?? 1
?? 2
+?? 2
=0,??? ?[0,1]
 
If we differentiate term-wise, we get 
??
?? ?? =1
??? ?? '
(?? )=?? ?? '
(?? )=
?? (1-?? 2
?? 2
)
(1+?? 2
?? 2
)
2
(??) 
At ?? =0 
??
?? ?? =1
?? ?? '
(?? )=?? ?? (0)=?? , which clearly does not tend to 0 as ?? ?8.  
????
?? ?? =1
?? ?? (?? )=?? ?? (?? )=
?? ?? 1+?? 2
?? 2
 can not be differentiated term by term at ?? =0.  
When ?? ?0, from (i) 
lim
?? ?8
???
?? ?? =1
?? ?? '
(?? )= lim
?? ?8
??? ?? '
(?? )=0= lim
?? ?8
???
?? ?? =1
?? ?? (?? ) 
Hence, ?
?? =1
8
??? ?? (?? ) can be differentiated term by term when ?? ?0. 
3.5 Show that if ?? (?? )=?
?? =?? 8
?
?? ?? ?? +?? ?? ?? ?? , then its derivative 
?? '
(?? )=-?? ?? ??
8
?? =?? ?? ?? ?? (?? +?? ?? ?? )
?? , for all ?? .  
(2011 : 20 Marks) 
Solution: 
Given : 
?? (?? )=??
8
?? =1
1
?? 3
+?? 4
?? 2
 
As 
1
?? 3
+?? 4
?? 2
=
1
?? 3
??? ??? ,?? ??? 
???? ?? =
1
?? 3
 and ??
8
?? =1
?? ?? =??
8
?? =1
1
?? 3
 converges.  
? By Weierstrass's M-test (A series of functions S?? ?? will converge uniformly and 
absolutely on [?? ,?? ] if there exists a convergent series S?? ?? of positive numbers such that 
for all ?? ?[?? ,?? ]. 
|?? ?? (?? )|=?? ?? for all ?? 
?? (?? )=??
8
?? =1
?? ?? (?? ) converges uniformly.  
Let 
?? ?? (?? )=?? ?? (?? )=
-2?? ?? 2
(1+?? ?? 2
)
2
 
and 
?? ?? '
(?? )=
-2+6?? ?? 2
?? 2
(1+?? ?? 2
)
3
 
For maximum and minimum, 
?? ?? '
(?? )=0 
??????????????????????????????????????????????????????????????-2+6?? ?? 2
=0??? 2
=
1
3?? 
It can be verified that ?? ?? ''
(?? )<0 for ?? 2
=
1
3?? .