Convergence or Divergence: Analyzing [(3^-n)+(n^-1)] Series from n=1 to inf

[BoldKnight399]

Ok, so the problem is [(3^-n)+(n^-1)] and I have to determine if it converges are diverges. from n=1 to inf. The problem is that individually I know that the 3^-n should converge and that n^-1 should diverge. But I don't understand what happens when your taking the series of the two combines. I think it would diverge, because of the n^-1, but I don;t know what test to prove it or if I even have the right idea. If anyone has any suggestions of a test to do, I tried the root test but found it to be 1 which is inconclusive, and I don't know where to go with this problem.

 

[Bohrok]

You know that 1/n diverges from the integral test, and that 3^-n is only making the sum bigger, so it diverges.

 

[BoldKnight399]

Thank you so much. I just wasn't sure if you could do that.

 

[statdad]

To firm this up, if a_n = e^{-n} + n^{-1} and b_n = n^{-1}.
you can use the comparison test to show that \sum_{n=1}^\infty a_n diverges.

 

[HallsofIvy]

By the way, it would have been clearer if you had stated from the start that the question was whether or not the series e^{-n}+n^{-1} converges. Of course, the sequence obviously converges to 0 so everyone assumed you mean "series" but it was ambiguous which you meant.