Series Convergence Tests

Convergent Series:

A series is convergent if the sequence of its partial sums converges. In more formal language, a series converges if there exists a limit l such that for any arbitrarily small positive number , there is a large integer N such that for all ,

Convergence Tests

Comparasion Test:

The terms of the sequence are compared to those of another one . If, for all n , , and converges, so does . However, if, for all n , , and diverges,so does .

Ratio Test:

Assume that for all n, a_n > 0. Suppose that there exists r such that

. If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

Root Test:

Suppose that the terms of the sequence in question are non-negative, and that there exists r such that

. If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

Convergence Tests Part II