Processing math: 100%

proof of slower divergent series


Let us show that, if the series i=1ai of positive terms is divergent, then Abel’s series

i=1aiij=1aj

also diverges.

Since the series i=1ai diverges, we can find an increasing sequence (ni)i=0 of integers such that

ni+1j=1aj>2nij=1aj

for all i. By convention, set n0=0. Then we can group the sum like so:

nmi=1aiij=1aj=m-1i=0ni-1k=ni+1akkj=1aj

Because kj=1ajni-1j=1aj, we have

ni-1k=ni+1akkj=1ajni-1k=ni+1akni-1j=1aj=ni-1k=ni+1akni-1j=1aj

By the way we chose the sequenceMathworldPlanetmath (ni)i=0, we have 2ni-1k=ni+1ak>ni-1j=1aj and, hence,

ni-1k=ni+1akkj=1ajni-1k=ni+1akni-1j=1aj>12.

Therefore,

nmi=1aiij=1aj>nmi=112=m2,

so the sum diverges in the limit m.

Title proof of slower divergent series
Canonical name ProofOfSlowerDivergentSeries
Date of creation 2013-03-22 15:08:42
Last modified on 2013-03-22 15:08:42
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Proof
Classification msc 40A05