I would like to prove the fact that the series of positive terms converges.
Precondition;
$\sum_{n=0}^{\infty} a_n$ is the series of positive terms, and
$\sum_{n=0}^{\infty} a_n$ converges.
Problem;
$\sum_{n=0}^{\infty} (a_n)^2$ converges.
I tried to prove that
$\sum_{k=0}^n a_k$ is Cauchy sequence $\implies \sum_{k=0}^n (a_k)^2$ is Cauchy sequence.
However, I couldn't prove this.
How can I prove that $\sum_{n=0}^{\infty} (a_n)^2$ converges?
Best Answer
Since $\sum_{n=0}^∞ a_n $ is convergent, we have $a_n \to 0$ , hence there is $N$ such that $a_n <1$ for $n >N$. Then we have
$$0<a_n^2 < a_n$$
for $n>N.$ Can you proceed ?