- 4 4 2.5² 3.5³ + 4 4.54 +
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Evaluate the following sum of the infinite series by recognizing it as either a transformation of a Taylor series or a Taylor series.
![The expression shown is an infinite series:
\[
\frac{4}{5} - \frac{4}{2 \cdot 5^2} + \frac{4}{3 \cdot 5^3} - \frac{4}{4 \cdot 5^4} + \cdots
\]
This series is an example of an alternating series, characterized by alternating positive and negative terms. Each term in the series follows a specific pattern where the numerator is "4" and the denominator increases following the formula \( n \cdot 5^n \), where "n" is the position of the term in the sequence starting from 1.
Key features of this series:
- The series begins with a positive term: \(\frac{4}{5}\).
- The subsequent term is subtracted: \(\frac{4}{2 \cdot 5^2}\).
- The alternating pattern of plus and minus continues.
The terms decrease in magnitude as n increases, due to the denominator's growth, suggesting the series may converge to a specific value as more terms are added.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb96d91a2-1904-4455-9ca0-61adba14ab53%2Ff8f3631c-6e78-4134-ba4c-1a44556beb3a%2Froscwjb_processed.png&w=3840&q=75)
Transcribed Image Text:The expression shown is an infinite series:
\[
\frac{4}{5} - \frac{4}{2 \cdot 5^2} + \frac{4}{3 \cdot 5^3} - \frac{4}{4 \cdot 5^4} + \cdots
\]
This series is an example of an alternating series, characterized by alternating positive and negative terms. Each term in the series follows a specific pattern where the numerator is "4" and the denominator increases following the formula \( n \cdot 5^n \), where "n" is the position of the term in the sequence starting from 1.
Key features of this series:
- The series begins with a positive term: \(\frac{4}{5}\).
- The subsequent term is subtracted: \(\frac{4}{2 \cdot 5^2}\).
- The alternating pattern of plus and minus continues.
The terms decrease in magnitude as n increases, due to the denominator's growth, suggesting the series may converge to a specific value as more terms are added.
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