Convergent Series -- From Wolfram MathWorld

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

Formally, the infinite series sum_(n=1)^(infty)a_n is convergent if the sequence of partial sums

S_n=sum_(k=1)^na_k (1)

is convergent. Conversely, a series is divergent if the sequence of partial sums is divergent. If sumu_k and sumv_k are convergent series, then sum(u_k+v_k) and sum(u_k-v_k) are convergent. If c!=0, then sumu_k and csumu_k 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

sum_(n=2)^(infty)1/(nlnn)=infty (2)
sum_(n=3)^(infty)1/(nlnn(lnlnn))=infty (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

sum_(n=2)^infty1/(n(lnn)^2) approx 2.109742801236 (4)

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

sum_(n=3)^infty1/(nlnn(lnlnn)^2) approx 38.4067680928... (5)

(OEIS A118582; Mathar 2009) converge by the integral test, although the latter converges so slowly that 10^(3.14×10^(86)) terms are needed to obtain two-digit accuracy (Zwillinger 1996, p. 39). Both can be summed using the Euler-Maclaurin integration formulas.

Tag » What Does It Mean When A Series Converges