# Integral test

### Integral test

In this section, we will learn about another test called the Integral test. The idea is to take the general term as a function in terms of x, and then integrate it. You can only use this test if the function is positively decreasing. If the integral gives a finite value, then the series is convergent. If the integral diverges to infinity, then the series is also divergent. We will first do some questions that require you to use the integral test instead of p-series test. Then we will use the integral test on a complicated series to see if it converges or diverges.

#### Lessons

Note *The integral test states the following:
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.
• Introduction
Integral Test Overview

• 1.
P Series versus Integral test
Use the integral test instead of the p-series test to show that the series converge or diverge.
a)
$\sum_{n=1}^{\infty}\frac{3}{n^2}$

b)
$\sum_{n=1}^{\infty}\frac{1}{n}$

• 2.
Convergence/Divergence of Integral Test
Determine whether the following series converge or diverge using the integral test.
a)
$\sum_{n=3}^{\infty}\frac{2}{(5n+4)^5}$

b)
$\sum_{n=1}^{\infty}\frac{1}{n^2+7n+12}$

• 3.
Advanced Question Regarding to the Integral Test
Determine if the series $\sum_{k=2}^{\infty}\frac{1}{k\ \ {^3}\sqrt{ln(4k)}}$ converges or diverges.