Integral Test

Theorem

Suppose ${\sum }_{n=1}^{\infty }{a}_n$ is a Series with positive terms $a_n$. Suppose both function $f$ and positive $N$ exists such that:

1. $f$ is continuous
2. $f$ is decreasing
3. $f(n)=a_n$ for all $n\ge N$
   Then: ${\sum }_{n=1}^{\infty }{a}_n\text{and}{\int }_N^{\infty }f(x)dx$
   both converge or both diverge.

Intuition: If you have a function that is continuous, decreasing and models the series for each term. Then if the integral converges / diverges so does the series.