Section 10.6

The Integral Test

If we replace the discrete with a continuous , we can use integration to determine if a sum converges.

1

Connecting to Integrals

Recall the Harmonic Series . We stated it diverges, but why?

Rectangles under a Curve

Imagine plotting the function . If we draw rectangles of width 1 starting at , their total area is exactly the sum of the series:

We know from Improper Integrals that . Since the rectangles "stick out" above the curve (Left Riemann Sum), their total area is larger than infinity. Thus, the series diverges.

2

The Integral Test

We can generalize this idea. If a function behaves like the series terms , they share the same fate.

Conditions

The function must be:

  • Continuous
  • Positive
  • Decreasing

(on the interval )

Theorem

Converges If:

Diverges If:

Reminder: The test tells you convergence, not the value. .

3

The p-Series Test

A Powerful Shortcut

Using the result of , we can instantly classify a huge family of series.

Converges
if
Diverges
if
4

Worked Examples

Example 1: Using the p-Series Test

Determine convergence of .

Rewrite with an exponent:

Here . Since , the series Diverges.

Example 2: Using the Integral Test

Test for convergence.

The function is positive, continuous, and decreasing for .


Since the integral diverges to infinity, the series Diverges.

5

Practice Quiz

Loading...