Does the Series \(\sum_{k=1}^{\infty}\frac{2+(-1)^k}{5^k}\) Converge?

[fishingspree2]

$$\sum_{k=1}^{\infty}\frac{2+\left(-1 \right)^{k}}{5^{k}}$$

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

 

[Billy Bob]

Break it into a sum of two series?

 

[Cyosis]

Since you're asking if there exists another test you could also use.

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

It converges absolutely if the limit is smaller than 1.

 

[fishingspree2]

Thank you, I have figured it out. I have another question if you don't mind

$$\sum_{k=1}^{\infty}\frac{\left|\cos k \right|}{k^{3}}$$

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

 

[VeeEight]

What values can |cos k| take? Perhaps consider a p-series