The nth-Term Divergence Test and Integral Test

Divergence Test and Integral Test

By the end of this lesson, you will be able to apply the Divergence Test and Integral Test to determine convergence or divergence.

The Divergence Test (nth-Term Test)

If $\lim_{n\to\infty} a_n \neq 0$, the series $\sum a_n$ diverges.

This is a necessary condition for convergence: if the terms don't go to zero, the sum can't converge.

Important: this test can only prove divergence, never convergence. If $\lim a_n = 0$, the test is inconclusive.

Example: $\sum \frac{n}{n+1}$: $\lim \frac{n}{n+1} = 1 \neq 0$ → diverges.

Harmonic series warning: $\sum 1/n$ has $\lim a_n = 0$ but diverges (tested by the integral test). The Divergence Test fails here.

The Integral Test

Setup: if $f(x)$ is positive, continuous, and decreasing on $[1, \infty)$ and $a_n = f(n)$, then:

$$\sum_{n=1}^\infty a_n \text{ converges} \iff \int_1^\infty f(x), dx \text{ converges}$$

The p-Series

$\sum_{n=1}^\infty \frac{1}{n^p}$ — apply the integral test using $\int_1^\infty x^{-p}, dx$:

$$\sum_{n=1}^\infty \frac{1}{n^p} \begin{cases}\text{converges} & p > 1\ \text{diverges} & p \leq 1\end{cases}$$

This is one of the most important results to memorize.

Examples:

• $\sum 1/n^2$ converges ($p = 2 > 1$)
• $\sum 1/n$ diverges ($p = 1$) — the harmonic series
• $\sum 1/\sqrt{n}$ diverges ($p = 1/2 < 1$)

Integral Test Example

Does $\sum_{n=1}^\infty \frac{1}{n^2+1}$ converge?

$f(x) = \frac{1}{x^2+1}$ — positive, continuous, decreasing for $x \geq 1$ ✓

$\int_1^\infty \frac{dx}{x^2+1} = [\arctan x]_1^\infty = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}$ (finite → converges)

So $\sum \frac{1}{n^2+1}$ converges (though not to $\pi/4$ — the integral test doesn't give the sum).

Summary

Divergence test: if $\lim a_n \neq 0$ → diverges. If $\lim a_n = 0$ → inconclusive.

Integral test: converts series convergence to improper integral convergence.

p-series: converges iff $p > 1$.