Is the Series \(\sum_{i=1}^{\infty} \ln(\cos(\frac{1}{n}))\) Convergent?

Thread starter TTob
Start date Jul 10, 2009
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Convergence Series Series convergence

In summary, series convergence is a mathematical concept that describes the behavior of an infinite sequence of numbers. There are multiple methods for determining convergence, including the ratio test, comparison test, integral test, and alternating series test. Absolute and conditional convergence refer to different types of convergence, with absolute convergence guaranteeing convergence and conditional convergence not. While there are no universal shortcuts or tricks for determining convergence, there are some general rules and methods that can help. If stuck, it can be helpful to try different methods and seek help from others.

Jul 10, 2009

TTob

Homework Statement

Check if the following series is convergent.
[tex]
\sum^{\infty}_{i=1}l n(cos(\frac{1}{n}))
[/tex]I have tried a lot of different tests without success.
I need some hint.

Thanks

Homework Equations

The Attempt at a Solution

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Jul 10, 2009

tiny-tim

Science Advisor

Homework Helper

Hi TTob!

Hint: this is ln of a product

Jul 10, 2009

g_edgar

Tayor's Theorem

1. What is a series convergence?

A series convergence is a mathematical concept that refers to the behavior of an infinite sequence of numbers. It describes whether the sum of the terms in the sequence approaches a finite number or if it diverges to infinity.

2. How do you determine if a series converges?

There are multiple methods for determining the convergence of a series. These include the ratio test, comparison test, integral test, and alternating series test. Each of these methods has its own set of conditions and criteria for determining convergence.

3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order in which the terms are added, whereas conditional convergence refers to a series that only converges when the terms are added in a specific order. In other words, absolute convergence guarantees convergence, while conditional convergence does not.

4. Are there any shortcuts or tricks for determining series convergence?

While there are no shortcuts or tricks that apply to all series, there are some general rules that can help determine convergence. For example, if a series has terms that approach zero as n approaches infinity, it is likely to converge. Additionally, if a series has alternating positive and negative terms, the alternating series test can be used to determine convergence.

5. What should I do if I'm stuck on determining series convergence?

If you are having trouble determining the convergence of a series, it can be helpful to try different methods and compare their results. Additionally, seeking help from a teacher, tutor, or online resources can provide valuable insights and tips on solving difficult series convergence problems.