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

1 Answer
Bill K.
Jul 31, 2015

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.

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