Find all the values of x such that the given series would converge. (x – 8)" - 8" n=1 The series is convergent from x = left end included (enter Y or N): to x = right end included (enter Y or N):
Find all the values of x such that the given series would converge. (x – 8)" - 8" n=1 The series is convergent from x = left end included (enter Y or N): to x = right end included (enter Y or N):
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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![**Problem Description:**
Find all the values of \( x \) such that the given series would converge.
\[
\sum_{n=1}^{\infty} \frac{(x - 8)^n}{8^n}
\]
The series is convergent:
- From \( x = \) [input box], left end included (enter Y or N): [input box]
- To \( x = \) [input box], right end included (enter Y or N): [input box]
**Explanation:**
The task is to determine the interval of convergence for the given infinite series. The series in question is a geometric series where the common ratio is determined by the expression \((x - 8)/8\). The convergence condition for a geometric series \(\sum a r^n\) is \(|r| < 1\). Therefore, you need to find the range of \(x\) such that this condition holds true, ensuring that the series converges.
Complete the input boxes with the appropriate values and indicate whether each endpoint is included in the interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2F1d2fac41-1727-4c20-a607-95ec2d56d301%2Fputq98l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Find all the values of \( x \) such that the given series would converge.
\[
\sum_{n=1}^{\infty} \frac{(x - 8)^n}{8^n}
\]
The series is convergent:
- From \( x = \) [input box], left end included (enter Y or N): [input box]
- To \( x = \) [input box], right end included (enter Y or N): [input box]
**Explanation:**
The task is to determine the interval of convergence for the given infinite series. The series in question is a geometric series where the common ratio is determined by the expression \((x - 8)/8\). The convergence condition for a geometric series \(\sum a r^n\) is \(|r| < 1\). Therefore, you need to find the range of \(x\) such that this condition holds true, ensuring that the series converges.
Complete the input boxes with the appropriate values and indicate whether each endpoint is included in the interval.
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