This question is asked already here
Proof verification: convergent series with a finite number of negative terms is Absolutely Convergent
But, answer to this question used different method (it does not explain how the Cauchy convergence criterion applied)
Let $s_n$ be nth partial sum of $\sum a_n$ and $t_n$ be nth partial sum of $\sum |a_n|$ and let $a_n≥0$ for all $n>K$ then,
if $m>n>K$ we have, $t_m-t_n= s_m-s_n$
Now, how to apply Cauchy convergence criterion to establish the convergence of $(t_n)$ ?
I know, by Cauchy convergence criterion we have, for given $\epsilon >0$ there exists $M(\epsilon)\in\mathbb{N}$ such that, if $m>n>M(\epsilon)$ then $|s_m-s_n|=|s_{n+1}+s_{n+2}+…+s_m|<\epsilon$
(But then, why should be this $M(\epsilon)>K$?)
how to proceed? Please help
Best Answer
Let $N(\epsilon)$ be an integer greater than both $M(\epsilon)$ and $K$. Then for all $m>n>N(\epsilon)$, $a_n, \ldots, a_m$ are all positive, and their sum is smaller than $\epsilon$. You may apply Cauchy’s criterion from here.