1 5. (a) Use the identity = 1/(z+ 1) – 1) to establish Σ 1 - > (z + 1)" |z + 1| < 1 n=0 (b) Use the above identity to also establish z2 (n + 1)(z + 1)" |z + 1| < 1 n=0 Verify that you get the same result by differentiation of the series in part (a).
1 5. (a) Use the identity = 1/(z+ 1) – 1) to establish Σ 1 - > (z + 1)" |z + 1| < 1 n=0 (b) Use the above identity to also establish z2 (n + 1)(z + 1)" |z + 1| < 1 n=0 Verify that you get the same result by differentiation of the series in part (a).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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need help with attached complex analysis problem dealing with series and proving, Thankyou in advance
![5. (a) Use the identity \( \frac{1}{z} = \frac{1}{(z + 1)} - 1 \) to establish
\[
\frac{1}{z} = -\sum_{n=0}^{\infty} (z + 1)^n \quad \text{for} \quad |z + 1| < 1
\]
(b) Use the above identity to also establish
\[
\frac{1}{z^2} = \sum_{n=0}^{\infty} (n + 1)(z + 1)^n \quad \text{for} \quad |z + 1| < 1
\]
Verify that you get the same result by differentiation of the series in part (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2Fc56b2e0b-5125-4b40-9e08-468688f38fd3%2F0z1vwn_processed.png&w=3840&q=75)
Transcribed Image Text:5. (a) Use the identity \( \frac{1}{z} = \frac{1}{(z + 1)} - 1 \) to establish
\[
\frac{1}{z} = -\sum_{n=0}^{\infty} (z + 1)^n \quad \text{for} \quad |z + 1| < 1
\]
(b) Use the above identity to also establish
\[
\frac{1}{z^2} = \sum_{n=0}^{\infty} (n + 1)(z + 1)^n \quad \text{for} \quad |z + 1| < 1
\]
Verify that you get the same result by differentiation of the series in part (a).
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