The Integral Test for a Positive Term Infinite Series

In a Nut Shell: For certain positive term series the integral test provides a quick method

to determine convergence or divergence of the series.

Integral test - If ∑ a_n is a positive series and f(x) is positive valued decreasing,

continuous function for x ≥ 1. If f(n) = a_n for all integers n ≥ 1, then the series and

the improper integral

∞

∑ a_n and ∫ f(x)dx

1

either both diverge or both converge. The integral must have a finite value for convergence.

Example: Use the integral test to determine if the following series converges or diverges.

∞

∑ 1 / (3n + 1)³

n = 1

Strategy: First check to see if conditions of the series are sufficient to apply the integral test.

The series is a positive term series.

The function, f(x) = 1 / (3x+1)³ , is a decreasing function

The function, f(x) = 1 / (3x+1)³ , is continuous.

They are satisfied.

∞

Evaluate the integral I = ∫ 1 / (3x + 1)³ Use the substitution u = 3x + 1

x = 1

t

So du = 3 dx and dx = (1/3) du and I = lim (1/3) ∫ u ^{˗ 3} du

t → ∞ 4

t

I = lim (1/3) [ u ^{˗ 2} /( ˗ 2) ] | = ( ˗ 1/6) lim [[ t ^{˗ 2} ˗ 4 ^{˗ 2} ]

t → ∞ 4 t → ∞

I = 16 / 6 = 8 / 3 (result)

Since the integral has a finite value the integral converges and so does the series. (result)

Return to Calculus 2 Notes Involving Sequences, Infinite Series, etc

Copyright © 2017 Richard C. Coddington

All rights reserved.