Calculus 50

Calculus

Lesson 50

Integral Test and p-series

For ANY series , we can generate a function such that

. Then, if , continuous, and decreasing for all ,

then and either both converge or both diverge.

Examples: diverges and converges

p-series is the same as the special form that we saw when we did indefinite

integrals.

converges when and diverges when

prove this with the integral test...

when , it is a special series called the "harmonic series."

We can also show that it diverges like this:

_____ ___________ _____________

>1/2 >1/2 >1/2

We know this since 2 (1/4s) would = 1/2 so 1/3 and 1/4 is more. 4 (1/8s) would be 1/2, and 1/5 and 1/6 and 1/7 and 1/8 is more, &c.

Practice: For each, determine the convergence:

Find the values of p for which converges.

Note for integral test: If the first few terms increase and then the

rest of them decrease, take the first few terms out, then

integrate from the earliest term in the new series to infinity.

Example:

On to Lesson 51 - Comparison Test