# Using Cauchy convergence criterion to prove that, "if convergent series contains only finitely many negative terms then it is absolutely convergent"

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 $$M(\epsilon)>K$$?)

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.