State the following. The Integral Test

The Integral Test states that: If $a_n = f(n)$ where $f$ is a continuous, positive, decreasing function on $[1, \infty)$. Then (i) $\sum_{1}^{\infty} a_n$ converges if and only if $\int_{1}^{\infty} f(x) dx$ converges, and (ii) $\sum_{1}^{\infty} a_n$ diverges if and only if $\int_{1}^{\infty} f(x) dx$ diverges.

$$\textbf{Integral Test}$$

Consider the following series

$$\sum_{n=k}^{\infty}f(n)$$

Where $f(x)$ is positive and monotone decreasing on $\left[k,\infty\right)$.

This series is convergent $\textbf{if and only if}$ the following integral is finite.

$$\int_k^{\infty}f(x)\hspace{1mm}dx$$