convergence tests

Divergence Test (nth term test)

If lim{n \to \infty} an \neq 0, then the series diverges. The test is inconclusive if lim{n \to \infty} an = 0.

Integral Test

If f(x) is continuous, positive, and decreasing on the interval [k, \infty), where k is some number, and an = f(n), then \sum{n=k}^{\infty} an and \int{k}^{\infty} f(x) dx either both converge or both diverge.

P-Series Test

A p-series is of the form \sum_{n=1}^{\infty} \frac{1}{n^p}. The p-series converges if p > 1 and diverges if p \le 1.

Comparison Test

If 0 \le an \le bn for all n, then:

• If \sum bn converges, then \sum an converges.
• If \sum an diverges, then \sum bn diverges.