Convergence of a Series

In summary, the conversation discusses the convergence of the series \sum_{k=1}^{\infty}\frac{2+\left(-1 \right)^{k}}{5^{k}}, with the person asking for suggestions on a test to use. Another person suggests breaking it into two series and using the limit comparison test. The original person then figures out that the series converges absolutely if \lim_{n \to \infty}\sqrt[n]{|a_n|}<1. They then ask for help with the series \sum_{k=1}^{\infty}\frac{\left|\cos k \right|}{k^{3}}, and someone suggests using a p-series test.
  • #1
[tex]\sum_{k=1}^{\infty}\frac{2+\left(-1 \right)^{k}}{5^{k}}[/tex]

Hello, I am trying to determine if this series converges or diverges. I have tried comparison test and d'Alembert's test but I was not successful

Can anyone suggest me a test learned in Calculus 2 that will work?

Thank you
 
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  • #2
Break it into a sum of two series?
 
  • #3
Since you're asking if there exists another test you could also use.

[tex]
\lim_{n \to \infty}\sqrt[n]{|a_n|}<1
[/tex]

It converges absolutely if the limit is smaller than 1.
 
  • #4
Thank you, I have figured it out. I have another question if you don't mind

[tex]\sum_{k=1}^{\infty}\frac{\left|\cos k \right|}{k^{3}}[/tex]

It looks like a riemann series but I don't really know how to deal with trigonometric functions in series yet. What to do? Can anyone explain me?

Thank you
 
  • #5
What values can |cos k| take? Perhaps consider a p-series
 

What is "Convergence of a Series"?

"Convergence of a Series" refers to the mathematical concept of determining whether a sequence of numbers, when summed together, will result in a finite or infinite value. This is important in many fields of science, including physics, engineering, and economics.

What is the difference between a convergent and a divergent series?

A convergent series is one in which the sequence of partial sums approaches a finite limit, meaning that the sum of the series is a finite value. On the other hand, a divergent series is one in which the sequence of partial sums does not approach a finite limit, and the sum of the series is either infinite or does not exist.

How is the convergence of a series tested?

There are various tests that can be used to determine the convergence of a series, such as the Ratio Test, the Root Test, and the Comparison Test. These tests involve examining the behavior of the terms in the series to determine whether they approach zero or not.

What is the significance of determining the convergence of a series?

Determining the convergence of a series is important in many applications, as it allows us to determine the behavior of a sequence of numbers when they are added together. This can help us understand the behavior of systems and make predictions about their future behavior.

Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. It must fall into one of these categories, depending on whether the sum of the series approaches a finite value or not. However, it is possible for a series to be conditionally convergent, meaning that it is convergent when certain conditions are met, but divergent otherwise.

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