sequence determining convergence of series

Theorem.  Let a1+a2+ be any series of real an.  If the positive numbers r1,r2,  are such that

limnanrn=L 0, (1)

then the series converges simultaneously with the series r1+r2+

Proof.  In the case that the limit (1) is positive, the supposition implies that there is an integer n0 such that

0.5L<anrn< 1.5Lfor nn0. (2)


0< 0.5Lrn<an< 1.5Lrnfor all nn0,

and since the series n=10.5Lrn and n=11.5Lrn converge simultaneously with the series r1+r2+, the comparison testMathworldPlanetmath guarantees that the same concerns the given series a1+a2+

The case where (1) is negative, whence we have


may be handled as above.

Note.  For the case  L=0, see the limit comparison testMathworldPlanetmath.

Title sequence determining convergence of series
Canonical name SequenceDeterminingConvergenceOfSeries
Date of creation 2013-03-22 19:06:54
Last modified on 2013-03-22 19:06:54
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Definition
Classification msc 40A05
Related topic LimitComparisonTest