1. (i) Use the Ratio Test to determine the open interval of convergence, Iº = (-R,R), and the radius of convergence, R, for the power series: (1 ots) Σ n-1 x" 2" n² (ii) Test each endpoint for convergence to find 7, the interval of convergence. (10p)) (iii) State the values of x where the series converges absolutely, converges conditionally or diverges. (
1. (i) Use the Ratio Test to determine the open interval of convergence, Iº = (-R,R), and the radius of convergence, R, for the power series: (1 ots) Σ n-1 x" 2" n² (ii) Test each endpoint for convergence to find 7, the interval of convergence. (10p)) (iii) State the values of x where the series converges absolutely, converges conditionally or diverges. (
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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![1.
(i) Use the Ratio Test to determine the open interval of convergence, \( I^o = (-R, R) \), and the radius of convergence, \( R \), for the power series:
\[
\sum_{n=1}^{\infty} \left( \frac{x^n}{2^n n^2} \right)
\]
(ii) Test each endpoint for convergence to find \( I \), the interval of convergence.
(iii) State the values of \( x \) where the series converges absolutely, converges conditionally, or diverges.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa06fb91f-1445-4bd6-b9dd-268642fc0d22%2F7ced479f-33d2-4e19-927c-9c69531d111c%2F02mbkqd.jpeg&w=3840&q=75)
Transcribed Image Text:1.
(i) Use the Ratio Test to determine the open interval of convergence, \( I^o = (-R, R) \), and the radius of convergence, \( R \), for the power series:
\[
\sum_{n=1}^{\infty} \left( \frac{x^n}{2^n n^2} \right)
\]
(ii) Test each endpoint for convergence to find \( I \), the interval of convergence.
(iii) State the values of \( x \) where the series converges absolutely, converges conditionally, or diverges.
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