Find whether sequence with general term $a_n=\sum_{i=1}^n \frac{1}{5^i+1}$ converges or diverges.

Question

Find whether the sequence whose general term is given by $$a_n= \frac{1}{5+1}+\frac{1}{5^2+1}+.....+ \frac{1}{5^n+1}$$ is convergent or divergent.

I compared this sequence with the sequence whose general term is

$$b_n=\frac{1}{5}+\frac{1}{5^2}+.....+ \frac{1}{5^n}$$

We know that $b_n $converges to $\frac{1}{4}$ and ${b_n} >{a_n}$, so $a_n$ will converge too.

Is my method correct? Thanks!