Series Convergence/Divergence Proofs

In summary, a series convergence/divergence proof is a mathematical method used to determine whether an infinite sum of terms will have a finite sum or continue to grow indefinitely. This is done through tests and techniques such as the ratio test, comparison test, integral test, and alternating series test. A series converges when the sum of its terms approaches a fixed value, while it diverges when the sum of its terms grows infinitely larger. A series cannot converge and diverge at the same time, and these proofs are important in areas such as calculus and numerical analysis to make accurate predictions and conclusions.
  • #1
chimychang
5
0
[tex] \sum_{n=1}^{\infty} n \sin(\frac{1}{n}) [/tex]

I rewrote the sum as [tex] \sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\frac{1}{n}} [/tex]

Then I applied the Nth term test and used L'Hoptials rule so [tex] \lim_{n\to\infty} \frac{\cos(\frac{1}{n})\frac{-1}{n^2}}{\frac{-1}{n^2}} [/tex]

The [tex] \frac{-1}{n^2} [/tex] cancel out and the [tex] lim_{n\to\infty} \cos(\frac{1}{n}) [/tex] is 1 which by the nth term test is divergent. Is that a legitimate proof of divergence?
 
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  • #2
Yes.
 

FAQ: Series Convergence/Divergence Proofs

What is a series convergence/divergence proof?

A series convergence/divergence proof is a mathematical method used to determine whether a series, which is an infinite sum of terms, will have a finite sum or will continue to grow indefinitely. This proof involves using specific tests and techniques to analyze the behavior of the series and determine its convergence or divergence.

What are some common tests used in series convergence/divergence proofs?

Some common tests used in series convergence/divergence proofs include the ratio test, the comparison test, the integral test, and the alternating series test. These tests are used to determine the behavior of a series and whether it converges or diverges.

How do you know when a series converges or diverges?

A series converges when the sum of its terms approaches a finite value as the number of terms increases. In other words, as more terms are added, the sum of the series gets closer and closer to a fixed value. On the other hand, a series diverges when the sum of its terms does not approach a finite value, but instead grows infinitely larger.

Can a series converge and diverge at the same time?

No, a series cannot converge and diverge at the same time. A series can only have one of these two outcomes. If a series fails to converge, it automatically diverges.

Why are series convergence/divergence proofs important?

Series convergence/divergence proofs are important because they allow us to determine the behavior of a series and whether it will have a finite sum or not. This information is crucial in many areas of mathematics, such as calculus and numerical analysis, and can help us make accurate predictions and conclusions based on the behavior of a series.

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