Compute the IM8 IM8 IM8 IM8 n=1 n=17 convergence set for the following power series. Use interval notation for your answers. xn (n + 1)2n (x - 1)" n (x - 1)n n! (x + 1)" 3n converges for converges for converges for converges for
Compute the IM8 IM8 IM8 IM8 n=1 n=17 convergence set for the following power series. Use interval notation for your answers. xn (n + 1)2n (x - 1)" n (x - 1)n n! (x + 1)" 3n converges for converges for converges for converges for
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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![**Power Series Convergence Analysis**
In this exercise, you are asked to compute the convergence set for the following power series. Please use interval notation for your answers:
1. \(\sum_{n=0}^{\infty} \frac{x^n}{(n+1)2^n}\) converges for __________
2. \(\sum_{n=1}^{\infty} \frac{(x-1)^n}{n}\) converges for __________
3. \(\sum_{n=1}^{\infty} \frac{(x-1)^n}{n!}\) converges for __________
4. \(\sum_{n=17}^{\infty} \frac{(x+1)^n}{3^n}\) converges for __________
### Explanation
To solve these problems, you will often use the Ratio Test for Convergence, defined as follows:
Given a series \(\sum a_n\), the Ratio Test states that the series converges if
\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1. \]
### Diagrams and Graphs
No diagrams or graphs are provided in this particular problem. The focus here is on determining the intervals of convergence using algebraic techniques.
### Steps for Solution
1. **Apply the Ratio Test**:
- For each series \(\sum a_n\), compute the limit of the ratio \(\left| \frac{a_{n+1}}{a_n} \right|\) as \(n\) approaches infinity.
- Solve for the values of \(x\) that make this limit less than 1.
2. **Determine the Interval of Convergence**:
- After solving for \(x\), specify the interval in interval notation.
3. **Check Endpoints (if necessary)**:
- This step involves testing the endpoints of the interval to see if the series converges at those specific points.
### Example
Let's consider the second power series for elaboration:
\[ \sum_{n=1}^{\infty} \frac{(x-1)^n}{n} \]
1. Identify \(a_n = \frac{(x-1)^n}{n}\).
2. Apply the Ratio Test:
\[
\lim_{n \to \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc5134d69-32d8-4659-bfc6-df6862919637%2F1155ed82-7014-4ff3-ac94-d6102108656f%2F9h9511j_processed.png&w=3840&q=75)
Transcribed Image Text:**Power Series Convergence Analysis**
In this exercise, you are asked to compute the convergence set for the following power series. Please use interval notation for your answers:
1. \(\sum_{n=0}^{\infty} \frac{x^n}{(n+1)2^n}\) converges for __________
2. \(\sum_{n=1}^{\infty} \frac{(x-1)^n}{n}\) converges for __________
3. \(\sum_{n=1}^{\infty} \frac{(x-1)^n}{n!}\) converges for __________
4. \(\sum_{n=17}^{\infty} \frac{(x+1)^n}{3^n}\) converges for __________
### Explanation
To solve these problems, you will often use the Ratio Test for Convergence, defined as follows:
Given a series \(\sum a_n\), the Ratio Test states that the series converges if
\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1. \]
### Diagrams and Graphs
No diagrams or graphs are provided in this particular problem. The focus here is on determining the intervals of convergence using algebraic techniques.
### Steps for Solution
1. **Apply the Ratio Test**:
- For each series \(\sum a_n\), compute the limit of the ratio \(\left| \frac{a_{n+1}}{a_n} \right|\) as \(n\) approaches infinity.
- Solve for the values of \(x\) that make this limit less than 1.
2. **Determine the Interval of Convergence**:
- After solving for \(x\), specify the interval in interval notation.
3. **Check Endpoints (if necessary)**:
- This step involves testing the endpoints of the interval to see if the series converges at those specific points.
### Example
Let's consider the second power series for elaboration:
\[ \sum_{n=1}^{\infty} \frac{(x-1)^n}{n} \]
1. Identify \(a_n = \frac{(x-1)^n}{n}\).
2. Apply the Ratio Test:
\[
\lim_{n \to \
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