1 2 (FF3 √k + 3 1 √k +2
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
Determine whether the series converges or diverges. If it diverges find the sum
![On this educational website, we will explore an infinite series involving nested square root functions.
The mathematical expression in question is:
\[ \sum_{k=1}^{\infty} \left( \frac{1}{\sqrt{k+3}} - \frac{1}{\sqrt{k+2}} \right) \]
Explanation:
- The symbol \(\sum\) denotes a summation.
- The limits of the summation are from \(k = 1\) to \(\infty\) (infinity), indicating that the summation runs over all natural numbers starting from 1 to infinity.
- Inside the summation is the expression \(\left( \frac{1}{\sqrt{k+3}} - \frac{1}{\sqrt{k+2}} \right)\).
Here, each term of the summation consists of the difference between two fractions:
- The first fraction is \(\frac{1}{\sqrt{k+3}}\), and
- The second fraction is \(\frac{1}{\sqrt{k+2}}\).
These terms involve square roots of expressions that increment in the denominator as \(k\) increases.
This sum is an example of a telescoping series, where many terms cancel each other out in the summation process, potentially simplifying the evaluation of the series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a0c2eed-fc4c-4fe4-9822-86e8a241e03b%2F98fd3448-0cd6-4808-a4a2-7b675eb5c8e7%2Fgjzuey_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On this educational website, we will explore an infinite series involving nested square root functions.
The mathematical expression in question is:
\[ \sum_{k=1}^{\infty} \left( \frac{1}{\sqrt{k+3}} - \frac{1}{\sqrt{k+2}} \right) \]
Explanation:
- The symbol \(\sum\) denotes a summation.
- The limits of the summation are from \(k = 1\) to \(\infty\) (infinity), indicating that the summation runs over all natural numbers starting from 1 to infinity.
- Inside the summation is the expression \(\left( \frac{1}{\sqrt{k+3}} - \frac{1}{\sqrt{k+2}} \right)\).
Here, each term of the summation consists of the difference between two fractions:
- The first fraction is \(\frac{1}{\sqrt{k+3}}\), and
- The second fraction is \(\frac{1}{\sqrt{k+2}}\).
These terms involve square roots of expressions that increment in the denominator as \(k\) increases.
This sum is an example of a telescoping series, where many terms cancel each other out in the summation process, potentially simplifying the evaluation of the series.
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