Find all values of x for which the series converges. (Enter your answer using interval notation.) 00 Σε | 8. n = 0 For these values of x, write the sum of the series as a function of x. f(x) %D
Find all values of x for which the series converges. (Enter your answer using interval notation.) 00 Σε | 8. n = 0 For these values of x, write the sum of the series as a function of x. f(x) %D
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 Statement
**Find all values of \(x\) for which the series converges. (Enter your answer using interval notation.)**
\[
\sum_{n=0}^{\infty} 8 \left( \frac{x - 5}{8} \right)^n
\]
\[ \text{Interval notation answer:} \quad [ \, \quad ] \]
**For these values of \(x\), write the sum of the series as a function of \(x\).**
\[
f(x) = \quad [ \, \quad ]
\]
### Explanation:
To determine where the series converges and to find its sum as a function of \(x\) for those values, the following steps should be followed:
1. **Identify the type of series:** The given series appears to be a geometric series of the form:
\[
\sum_{n=0}^{\infty} ar^n
\]
where \(a = 8\) and \(r = \frac{x - 5}{8}\).
2. **Convergence of a geometric series:** A geometric series converges if the absolute value of the common ratio \(r\) is less than 1:
\[
|r| < 1
\]
3. **Apply the convergence condition:** Substitute \(r\) with \(\frac{x - 5}{8}\):
\[
\left|\frac{x - 5}{8}\right| < 1
\]
This inequality can be solved to find the interval of \(x\) values for which the series converges.
4. **Summing the series:** If the series converges, its sum can be found using the formula for the sum of an infinite geometric series:
\[
S = \frac{a}{1 - r}
\]
Substitute \(a = 8\) and \(r = \frac{x - 5}{8}\) into this formula to find the sum as a function of \(x\).
### Details and Interval Notation
We'll solve:
\[
\left|\frac{x - 5}{8}\right| < 1
\]
This implies:
\[
-1 < \frac{x - 5}{8} < 1
\]
Multiplying through by 8:
\[
-8 < x - 5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F795f20db-de84-4e64-b0fe-ecefbfde6505%2Fea3d3843-3ace-4639-8ff0-7ce885cec962%2Fhqc15h7.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Find all values of \(x\) for which the series converges. (Enter your answer using interval notation.)**
\[
\sum_{n=0}^{\infty} 8 \left( \frac{x - 5}{8} \right)^n
\]
\[ \text{Interval notation answer:} \quad [ \, \quad ] \]
**For these values of \(x\), write the sum of the series as a function of \(x\).**
\[
f(x) = \quad [ \, \quad ]
\]
### Explanation:
To determine where the series converges and to find its sum as a function of \(x\) for those values, the following steps should be followed:
1. **Identify the type of series:** The given series appears to be a geometric series of the form:
\[
\sum_{n=0}^{\infty} ar^n
\]
where \(a = 8\) and \(r = \frac{x - 5}{8}\).
2. **Convergence of a geometric series:** A geometric series converges if the absolute value of the common ratio \(r\) is less than 1:
\[
|r| < 1
\]
3. **Apply the convergence condition:** Substitute \(r\) with \(\frac{x - 5}{8}\):
\[
\left|\frac{x - 5}{8}\right| < 1
\]
This inequality can be solved to find the interval of \(x\) values for which the series converges.
4. **Summing the series:** If the series converges, its sum can be found using the formula for the sum of an infinite geometric series:
\[
S = \frac{a}{1 - r}
\]
Substitute \(a = 8\) and \(r = \frac{x - 5}{8}\) into this formula to find the sum as a function of \(x\).
### Details and Interval Notation
We'll solve:
\[
\left|\frac{x - 5}{8}\right| < 1
\]
This implies:
\[
-1 < \frac{x - 5}{8} < 1
\]
Multiplying through by 8:
\[
-8 < x - 5
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