The general term is
[tex]a_n=\dfrac{n}{4^n}[/tex]
In order to apply the ratio test, we have to compute the ratio between two consecutive terms:
[tex]\dfrac{a_{n+1}}{a_n}=\dfrac{\frac{n+1}{4^{n+1}}}{\frac{n}{4^n}}=\dfrac{4^n(n+1)}{4^{n+1}n}=\dfrac{4^n(n+1)}{4^{n}\cdot 4n}=\dfrac{n+1}{4n}[/tex]
The limit as [tex]n\to\infty[/tex] of this ratio is
[tex]\displaystyle \lim_{n\to\infty} \dfrac{n+1}{4n}=\dfrac{1}{4}[/tex]
The ratio test states that if this ratio is between -1 and 1 (excluded), the series converges.
So, this series converges.
