Question

How do you test the series `(n / (5^n) )` from n = 1 to infinity for convergence?

Answer 1

Probably the best way is to use the Ratio Test to see that the series `sum_{n=1}^{infty}n/(5^(n))` converges.

Explanation:

Let `a_{n}=n/(5^(n))`. If `lim_{n->infty}|a_{n+1}|/|a_{n}| < 1`, the Ratio Test will imply that `sum_{n=1}^{\infty}a_{n}=sum_{n=1}^{infty}n/(5^(n))` converges.

Now `a_{n+1}=(n+1)/(5^(n+1))=(n+1)/(5*5^(n))`. Therefore,
since these terms are positive,

`|a_{n+1}|/|a_{n}|=((n+1)/(5*5^(n)))/(n/(5^(n)))=(n+1)/(5n)->1/5` as `n->infty`.

Hence,
`sum_{n=1}^{\infty}a_{n}=sum_{n=1}^{infty}n/(5^(n))` converges.