The tutor shows a simple example of the integral test.

The integral test for convergence of an infinite series states that, when a_n = f(n) both

∑a_n and ∫₁^∞f(x)dx

either converge or diverge.

We can therefore use the integral test to decide whether an infinite series converges or diverges.

Example: Does the infinite series

∑₁^∞(1/x)^3/2

converge?

Solution:

Using the integral test, we have

∫₁^∞x^-3/2 = -2x^-1/2|₁^∞ = 0- -2=2

Since the integral converges, so does the series.

HTH:)

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus. Toronto:
DC Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.