Topics: Calculus - Sequence - Series

The convergence of series can be tested with the following tests:

Sequence Converging to 0

(theorem, condición del resto)

Let be a convergent sequence, then:

This tells us that if the sequence that corresponds to a series doesn’t converge to , then the series diverges.

Comparison Test


Suppose that , and that converges.

Then, converges, and .

Comparison Test 2


Let and be series, and . Furthermore, .

Then converges if and only if converges.

Ratio Test


Let and .

Then, converges if and diverges if .

Integral Test


Let be a sequence and a non-negative decreasing function, such that .

Then, (an improper integral) converges if and only if converges.