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 $|a_n| < 10^{-n}$, prove that $\sum^{\infty}_{n=1} a_n$ converges.
Could someone give me a hint as to how to start this?
Hint If $|a_n|<10^{-n}$, then $\sum|a_n|<\sum 10^{-n} <\infty$
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.