●● (−1)n 2 n
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
considering the following series;
Give a number C such that you can guarantee that the sum of this series is no more than C
![The image displays a mathematical series expressed by the following formula:
\[ \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \]
This represents an infinite series where the summation is taken from \( n = 1 \) to infinity. The general term of the series is \(\frac{(-1)^n}{n^2}\), which alternates in sign based on the exponent \( n \). Each term's numerator alternates between 1 and -1, divided by the square of \( n \).
Explanation:
- The Greek capital letter sigma (Σ) denotes summation.
- The term \( n=1 \) at the bottom of the sigma indicates that the summation starts from \( n = 1 \).
- The infinity symbol (∞) at the top of the sigma indicates that the summation continues indefinitely.
- Inside the summation, \( \frac{(-1)^n}{n^2} \) indicates the formula to be summed.
This series is used in various areas of mathematical analysis and can be relevant when studying series convergence and alternating series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3c46422-8c7d-49ee-8251-70b5bf209c8b%2F8cf7e74c-27bf-4156-a6a0-6053427ff8e3%2Fpmwgrtj_processed.png&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical series expressed by the following formula:
\[ \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \]
This represents an infinite series where the summation is taken from \( n = 1 \) to infinity. The general term of the series is \(\frac{(-1)^n}{n^2}\), which alternates in sign based on the exponent \( n \). Each term's numerator alternates between 1 and -1, divided by the square of \( n \).
Explanation:
- The Greek capital letter sigma (Σ) denotes summation.
- The term \( n=1 \) at the bottom of the sigma indicates that the summation starts from \( n = 1 \).
- The infinity symbol (∞) at the top of the sigma indicates that the summation continues indefinitely.
- Inside the summation, \( \frac{(-1)^n}{n^2} \) indicates the formula to be summed.
This series is used in various areas of mathematical analysis and can be relevant when studying series convergence and alternating series.
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