Let r = - For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise. A. Consider the sequence {nr"}. lim nr" = 0 121-400 B. Take my word for it that it can be shown that Now consider the series ∞ n=1 nr." MINF n=1 nr" i=1 ir¹ - nr+2 (n+1)+1+r (1-r)²
Let r = - For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise. A. Consider the sequence {nr"}. lim nr" = 0 121-400 B. Take my word for it that it can be shown that Now consider the series ∞ n=1 nr." MINF n=1 nr" i=1 ir¹ - nr+2 (n+1)+1+r (1-r)²
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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![### Explanation of Convergence and Divergence
**Given:** \( r = \frac{17}{24} \).
When examining the convergence or divergence of sequences and series, follow these guidelines:
1. **Convergence and Divergence Definitions:**
- A sequence or series converges if it approaches a specific value.
- A sequence or series diverges if it increases or decreases without bound.
- For a sequence, provide the limit if convergent.
- For a series, provide the sum if convergent.
- Use `INF` if it diverges to infinity, `MINF` if it diverges to minus infinity, or `DIV` otherwise.
### Problem A: Sequence Analysis
**Sequence Considered:** \(\{nr^n\}\).
- **Limit Evaluation:**
\[
\lim_{n \to \infty} nr^n = 0
\]
The sequence converges to 0.
### Problem B: Series Analysis
**Given Identity:**
\[
\sum_{i=1}^{n} ir^i = \frac{nr^{n+2} - (n+1)r^{n+1} + r}{(1-r)^2}
\]
- **Series Considered:**
\[
\sum_{n=1}^{\infty} nr^n
\]
The series diverges to minus infinity, noted as `MINF`.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccac55ee-732b-4607-a38e-f5bd0a8aacb2%2F49d1f02f-32fb-4756-b372-f8673cf53199%2F2210od_processed.png&w=3840&q=75)
Transcribed Image Text:### Explanation of Convergence and Divergence
**Given:** \( r = \frac{17}{24} \).
When examining the convergence or divergence of sequences and series, follow these guidelines:
1. **Convergence and Divergence Definitions:**
- A sequence or series converges if it approaches a specific value.
- A sequence or series diverges if it increases or decreases without bound.
- For a sequence, provide the limit if convergent.
- For a series, provide the sum if convergent.
- Use `INF` if it diverges to infinity, `MINF` if it diverges to minus infinity, or `DIV` otherwise.
### Problem A: Sequence Analysis
**Sequence Considered:** \(\{nr^n\}\).
- **Limit Evaluation:**
\[
\lim_{n \to \infty} nr^n = 0
\]
The sequence converges to 0.
### Problem B: Series Analysis
**Given Identity:**
\[
\sum_{i=1}^{n} ir^i = \frac{nr^{n+2} - (n+1)r^{n+1} + r}{(1-r)^2}
\]
- **Series Considered:**
\[
\sum_{n=1}^{\infty} nr^n
\]
The series diverges to minus infinity, noted as `MINF`.
![Let \( r = \frac{17}{24} \).
For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise.
A. Consider the sequence \(\{nr^n\}\).
\[
\lim_{n \to \infty} nr^n = 0
\]
B. Take my word for it that it can be shown that
\[
\sum_{i=1}^{n} ir^i = \frac{nr^{n+2} - (n+1)r^{n+1} + r}{(1-r)^2}.
\]
Now consider the series \(\sum_{n=1}^{\infty} nr^n\).
\[
\sum_{n=1}^{\infty} nr^n = \text{INF}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccac55ee-732b-4607-a38e-f5bd0a8aacb2%2F49d1f02f-32fb-4756-b372-f8673cf53199%2Fx494czr_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( r = \frac{17}{24} \).
For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise.
A. Consider the sequence \(\{nr^n\}\).
\[
\lim_{n \to \infty} nr^n = 0
\]
B. Take my word for it that it can be shown that
\[
\sum_{i=1}^{n} ir^i = \frac{nr^{n+2} - (n+1)r^{n+1} + r}{(1-r)^2}.
\]
Now consider the series \(\sum_{n=1}^{\infty} nr^n\).
\[
\sum_{n=1}^{\infty} nr^n = \text{INF}
\]
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