Question

Is the series `\sum_(n=1)^\infty((-5)^(2n))/(n^2 9^n)` absolutely convergent, conditionally convergent or divergent?

(Use the appropriate test)

Answer 1

Diverges by the Ratio Test.

Explanation:

We'll check for Absolute Convergence using the Ratio Test, where `a_n=(-5)^(2n)/(n^2 9^n)`.

The Ratio Test tells us we take

`L=lim_(n->oo)|a_(n+1)/a_n|,` and

`L<1` implies absolute convergence (and hence convergence),

`L>1` implies divergence,

`L=1` is inconclusive and we need to use another test.

Then,

`a_(n+1)=(-5)^(2(n+1))/((n+1)^2 9^(n+1))=(-5)^(2n+2)/((n+1)^2 9^(n+1))`

Thus,

`L=lim_(n->oo)|(-5)^(2n+2)/((n+1)^2 9^(n+1))*(n^2 9^n)/(-5)^(2n)|`

`(-5)^(2n+2)/(-5)^(2n)=(-5)^2=25`

`9^n/9^(n+1)=1/9`

We can factor these constants outside of the limit, getting

`25/9lim_(n->oo)n^2/(n+1)^2=25/9>1`

We dropped the absolute value bars as everything is positive when we go to infinity.

We also then have divergence by the Ratio Test.