Begin2.DVI

Example 12-1. (Sum of series)

Find the sum of the infinite series

1

1

−^1

4

+^1

7

−^1

10

+^1

13

−^1

16

+··· (12.7)

which is an example of the series (12.2) when a= 1 and b= 3.

Solution

Use the equation (12.6) to find the sum of the given infinite series by evaluating

the integral

I=

∫ 1

0

dt

1 + t^3 (12.8)

Use partial fractions and write

1

1 + t^3

= A

1 + t

+ Bt +C

1 −t+t^2

and show that A= 1 / 3 ,B=− 1 / 3 and C= 2 / 3. Using some algebra and completing

the square on the denominator term the required integral can be reduced to the

following standard forms

I=

∫ 1

0

dt

1 + t^3

=^1

3

∫ 1

0

dt

1 + t

+

∫ 1

0

−t/3 + 2 / 3

1 −t+t^2

dt

=^1

3

∫ 1

0

dt

1 + t

−^1

6

∫ 1

0

2 t− 1

1 −t+t^2

dt +^1

2

∫ 1

0

dt

(√ 3

2

) 2

+ (t− 1 /2)^2

where each integral is in a standard form which can be easily integrated. If you don’t

recognize these integrals then look them up in an integration table. Integration of

each term produces

I=

[

√^1

3

tan −^1

(

2 t√− 1

3

)

+^1

3

ln(1 + t)−^1

6

ln(1 −t+t^2 )

] 1

0

=^1

3

(

√π

3

+ ln(2)

)

(12.9)

giving the final result

I=^1

1

−^1

4

+^1

7

−^1

10

+^1

13

−^1

16

+··· =^1

3

(

√π

3

+ ln(2)

)

(12 .10)