T**he Integral Test**

Suppose (1): f(x) is a continuous, positive, decreasing function on [1, ∞) and (2): let an = f(n).

Then the Infinite Series an from n=1 to ∞ n=1 an is convergent if and only if the improper integral goes to a Real number , L and Integral of f(x) dx from 1 to infinity is convergent.

Integral Test and Divergence/Convergence of series :

Assume conditions (1) and (2) of Integral Test.

• If Integral of f(x) from 1 to infinity is convergent, then the Infinite Series from n=1 to infinity is convergent.

• If Integral of f(x) from 1 to infinity is divergent, then the Infinite Series from n=1 to infinity is divergent.

* The p-series Definition:

Let p be a real number. Then the infinite series from n=1 to ∞ of 1/n ^ p is called the p-series.

**Theorem: The p-series is convergent if p > 1 and divergent if p ≤ 1.