00 If f(2) = Ë * 00 a"* and g(a) = Ï ( – 1)"", (e) – g(=)). a", find the power series of n=0 n=0
00 If f(2) = Ë * 00 a"* and g(a) = Ï ( – 1)"", (e) – g(=)). a", find the power series of n=0 n=0
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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![If \( f(x) = \sum_{n=0}^{\infty} \frac{n}{3^n} x^n \) and \( g(x) = \sum_{n=0}^{\infty} (-1)^n \frac{n}{3^n} x^n \), find the power series of \( \frac{1}{2} \left( f(x) - g(x) \right) \).
\[
\sum_{n=0}^{\infty}
\]
### Explanation:
The problem involves finding a new power series that derives from two given power series, \( f(x) \) and \( g(x) \). The function \( f(x) \) is an infinite series involving a factor of \( \frac{n}{3^n} x^n \), whereas \( g(x) \) adds an alternating sign factor \( (-1)^n \).
To find the power series of \( \frac{1}{2} (f(x) - g(x)) \), you need to subtract the series for \( g(x) \) from that of \( f(x) \), resulting in another power series, and then multiply by \( \frac{1}{2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08a1092a-53fa-45f1-8cc5-326ab3991212%2F01beb02c-915e-4d64-9b9e-68cffcaad503%2Fiinocy_processed.png&w=3840&q=75)
Transcribed Image Text:If \( f(x) = \sum_{n=0}^{\infty} \frac{n}{3^n} x^n \) and \( g(x) = \sum_{n=0}^{\infty} (-1)^n \frac{n}{3^n} x^n \), find the power series of \( \frac{1}{2} \left( f(x) - g(x) \right) \).
\[
\sum_{n=0}^{\infty}
\]
### Explanation:
The problem involves finding a new power series that derives from two given power series, \( f(x) \) and \( g(x) \). The function \( f(x) \) is an infinite series involving a factor of \( \frac{n}{3^n} x^n \), whereas \( g(x) \) adds an alternating sign factor \( (-1)^n \).
To find the power series of \( \frac{1}{2} (f(x) - g(x)) \), you need to subtract the series for \( g(x) \) from that of \( f(x) \), resulting in another power series, and then multiply by \( \frac{1}{2} \).
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