Calculus-Sec-9-5-Solution

9.5   Taylor, Maclaurin, and Binomial Series     by SGLee - HSKim - JHLee

1-5. Determine the radius of convergence of the Maclaurin series expansion for , where

1.

,

,

,

,

,

.

Therefore, the function’s Maclaurin series is

.

Here , then   as .

Hence, by the Ratio Test, the series converges for all , and the radius of convergence is .

2.

for all . Therefore, Maclaurin series is

.

Let , then  as .

Hence by the Ratio Test, the series converges for all , and the radius of convergence is .

3.

,

,

,

,

,

.

Therefore, its Maclaurin series is

.

Here , then  as .

Hence by the Ratio Test, the series converge when .

4.

.

Therefore, its Maclaurin series is

.

Let , then  as .

Hence by the Ratio Test, the series converges for all , and the radius of convergence is .

6-9. Obtain the Taylor series for  about , where

6.

.

7.

10-12. Find the Maclaurin series for the given function.

10.  (Use )

.

Thus .

13-14. Evaluate the indefinite integral as an infinite series

13.

Integrate both sides term by term:

Thus

Integrate both sides term by term:

.

15-16. Evaluate the limit using a series:

15.

Ans:

16.

.

.

.

17. Deduce from the Maclaurin series for  that .

Since  ,

.

Put , then .

18-21. Obtain the binomial series and radius of convergence of the function.

18.

this series converges when , so the radius of convergence is .

19.

this series converges when . so the radius of convergence is .

20.

this series converges when , so the radius of convergence is .

21.

this series converges when , so the radius of convergence is .

22. Evaluate  using the binomial series where .

Since  ,

.

Thus ,

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