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. 
http://matrix.skku.ac.kr/cal-lab/cal-10-5-14.html
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 
,
Mathematicians Playing Cards:
http://matrix.skku.ac.kr/2009-Album/2009-Math-Poster2/2009-Math-Poster2.html
8. 
, 
9. 
, 
11. 
12. 
14. 
