Integral Test

Main theorem and definition:

Suppose f (x) is a function and you want to know if converges.

Check three conditions:

1. Is f (x) continuous?

2. Is f (x) always positive?

3. Is f (x) decreasing? (i.e., is f (x) negative?)

Then if f (x)dx converges, then converges.

And if f (x)dx diverges, then diverges.

Definition: A p-series is a series of the form . Using the Integral Test, we can show that the series converges if p > 1 and diverges to infinity if p ≤ 1.

Hints and tips:

• It’s important to remember that the Integral Test only applies to series with positive terms. However, after you learn about absolute convergence later, you may be able to use it for series with some negative terms by taking their absolute value and seeing if they are absolutely convergent.

• The Integral Test is one of the few two-way tests − you can use it to conclude that a series converges or that it diverges.

• Don’t make the common mistake of confusing p-series with geometric series. p-series has the n in the base and a constant exponent. Geometric series has a constant base and n in the exponent. When a geoemetric series converges, we have formulas to tell us what it converges to. When a p-series converges, we usually don’t know what it converges to.

• It’s worth memorizing the rule for p-series, since these are often used later in conjunction with the Comparison Test.

• Remember that the Integral Test never tells you exactly what a series converges to.