Convergent Series

A series is said to be convergent if it approaches some limit (D'Angelo and West 2000, p. 259).

Formally, the infinite series is convergent if the sequence of partial sums

(1)

is convergent. Conversely, a series is divergent if the sequence of partial sums is divergent. If and are convergent series, then and are convergent. If , then and both converge or both diverge. Convergence and divergence are unaffected by deleting a finite number of terms from the beginning of a series. Constant terms in the denominator of a sequence can usually be deleted without affecting convergence. All but the highest power terms in polynomials can usually be deleted in both numerator and denominator of a series without affecting convergence.

If the series formed by taking the absolute values of its terms converges (in which case it is said to be absolutely convergent), then the original series converges.

Conditions for convergence of a series can be determined in the Wolfram Language using SumConvergence[a, n].

The series

			(2)
			(3)

both diverge by the integral test, although the latter requires a googolplex number of terms before the partial sums exceed 10 (Zwillinger 1996, p. 39). In contrast, the sums

(4)

(Baxley 1992; Braden 1992; Zwillinger 1996, p. 39; Kreminski 1997; OEIS A115563) and

(5)

(OEIS A118582; Mathar 2009) converge by the integral test, although the latter converges so slowly that terms are needed to obtain two-digit accuracy (Zwillinger 1996, p. 39). Both can be summed using the Euler-Maclaurin integration formulas.