Does the Series $\sum\frac{2}{2^{n}}$ Converge or Diverge?

[asset101]

Does the series \sum\frac{2}{2^{n}}


converge??

(Note that bounds on the running index n are from 1 to infinity)

I have tried ratio test but it returned a value of 1 (showing nothing).

I can see from the table function on my calulator that the term eventually diminsh to (not including zero).

Any help would be appreciated.

 

[HallsofIvy]

Can you show how you did the ratio test? The ratio is NOT 1.

 

[AUMathTutor]

Rememer, you can pull anything not involving an "n" out of the summation. So the 2 in the numerator can come out. Then you have

2 (SUM) 1/2^n

This can also be written as

2 (SUM) (1/2)^n

You should be able to recognize the SUM as a notable one, and use the rules pertaining to that kind of summation to determine whether or not it converges. And if it converges, does multiplying by 2 change any of that?

 

[asset101]

Sorry fellas the question should read

Does the series \sum\frac{n}{2^{n}}


converge??

(Note that bounds on the running index n are from 1 to infinity)

I have tried ratio test but it returned a value of 1 (showing nothing).

I can see from the table function on my calulator that the term eventually diminsh to (not including) zero.

Any help would be appreciated

 

[Dick]

Ratio test should still not give you a value of 1. Can you show how you got that?

 

[nicksauce]

What comes to my mind would be doing a comparison test with the integral of x*exp(-x).

 

[Mark44]

What Dick said. The ratio test is definitely the test to use. If you got a limit of 1, you're doing something wrong.