If f is Decreasing, Continuous, and Positive for $x\ge1$ AND $a_n=f(x)$, then $\sum_{n=1}^\infin a_n$ and $\int_1^\infin f(x)dx$ either BOTH converge or diverge.

• Note 1:This does NOT mean that the series converges to the value of the definite integral!
• Note 2:The function need only be decreasing for all x >k for some k >1. ←Overall

$Error$

• If the series converges to $S$, then the remainder, $R_n=|S-S_n|$ is bounded by

$0\le R_n \le \int_n^\infin f(x)dx$. This means $S\in「S_n,S_n+R_n].$

• reference and proofs