Convergence of an Infinite Series involving Absolute Values If $|a_n| < 10^{-n}$, prove that $\sum^{\infty}_{n=1} a_n$ converges.
Could someone  give me a hint as to how to start this?

If $A_n = \sum^{n}_{i=1} a_i$,
if $n > m$,
$$A_n - A_m = \sum^{n}_{i=m+1} a_i < \sum^{n}_{i=m+1} 10^{-i} < 10^{-m}$$
(you can do better, but this is enough).
Then apply the Cauchy criterion.
