(2x+1)" n2 5. n=1 (> Answer (v Solution Use the ratio test : (2æ + 1)"+'/(n + 1)²| lim n2 lim n+00 n2 + 2n + 1 | 2x + 1| = |2x + 1|. |(2x + 1)"/n²| n00 Therefore the series converges absolutely for |2x +1|< 1, or |æ +1/2|< 1/2. The radius of convergence is p = 1/2. At x = 0 and x = -1, the series also converges absolutely.
(2x+1)" n2 5. n=1 (> Answer (v Solution Use the ratio test : (2æ + 1)"+'/(n + 1)²| lim n2 lim n+00 n2 + 2n + 1 | 2x + 1| = |2x + 1|. |(2x + 1)"/n²| n00 Therefore the series converges absolutely for |2x +1|< 1, or |æ +1/2|< 1/2. The radius of convergence is p = 1/2. At x = 0 and x = -1, the series also converges absolutely.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
The question is asking to determine the radius of convergence of the given power series. But I did not understand what the textbook answer meant by discussion about x=2 and x=4
Transcribed Image Text:(2x+1)"
5.
n2
n=1
Answer
Solution
Use the ratio test :
(2æ + 1)"+"/(n + 1)
lim
n?
|(2x + 1)"/n²|
lim
n→0 n2 + 2n + 1
| 2x + 1| = |2x + 1|.
Therefore the series converges absolutely for |2x + 1| < 1, or |x + 1/2|< 1/2. The radius of convergence is p =1/2. At x = 0 and x = –1, the
series also converges absolutely.
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