If $f(x)=a_n$ and $f(x)$ is a continuous, positive decreasing function from $[i,\infty]$, then we can say that:

1. If $\int_{i}^{\infty}f(x)dx$ is convergent, then the series $\sum_{n=i}^{\infty}a_n$ is also convergent.

2. If $\int_{i}^{\infty}f(x)dx$ is divergent, then the series $\sum_{n=i}^{\infty}a_n$ is also divergent.