-2)"x" n 5. n=1
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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Question
Determine the radius of convergence and the interval of convergence for each power series.
![### Summation Problem
**Problem 5: Evaluate the Series**
Evaluate the infinite series given by:
\[
\sum_{n=1}^{\infty} \frac{(-2)^n x^n}{\sqrt[4]{n}}
\]
**Description:**
This summation represents an infinite series where \( n \) starts at 1 and approaches infinity. The general term of the series is
\[
\frac{(-2)^n x^n}{\sqrt[4]{n}}
\]
where:
- \( (-2)^n \) denotes the base -2 raised to the power of \( n \).
- \( x^n \) signifies \( x \) raised to the \( n \)-th power.
- \( \sqrt[4]{n} \) is the fourth root of \( n \).
The task is to evaluate this series and assess its convergence properties. Understanding the behavior of this series can involve techniques such as the comparison test, ratio test, or root test, among others, depending on the context and complexity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41847e25-11eb-4a46-b15a-a716d47a2bff%2F79e1fdee-5c33-4f8d-bdcf-f32b3f501a63%2Fq9pjw0m_processed.png&w=3840&q=75)
Transcribed Image Text:### Summation Problem
**Problem 5: Evaluate the Series**
Evaluate the infinite series given by:
\[
\sum_{n=1}^{\infty} \frac{(-2)^n x^n}{\sqrt[4]{n}}
\]
**Description:**
This summation represents an infinite series where \( n \) starts at 1 and approaches infinity. The general term of the series is
\[
\frac{(-2)^n x^n}{\sqrt[4]{n}}
\]
where:
- \( (-2)^n \) denotes the base -2 raised to the power of \( n \).
- \( x^n \) signifies \( x \) raised to the \( n \)-th power.
- \( \sqrt[4]{n} \) is the fourth root of \( n \).
The task is to evaluate this series and assess its convergence properties. Understanding the behavior of this series can involve techniques such as the comparison test, ratio test, or root test, among others, depending on the context and complexity.
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