proof of slower divergent series
Let us show that, if the series ∑∞i=1ai of positive terms is divergent, then Abel’s series
∞∑i=1ai∑ij=1aj |
also diverges.
Since the series ∑∞i=1ai diverges, we can find an increasing sequence (ni)∞i=0 of integers such that
ni+1∑j=1aj>2ni∑j=1aj |
for all i. By convention, set n0=0. Then we can group the sum like so:
nm∑i=1ai∑ij=1aj=m-1∑i=0ni-1∑k=ni+1ak∑kj=1aj |
Because ∑kj=1aj≤∑ni-1j=1aj, we have
ni-1∑k=ni+1ak∑kj=1aj≥ni-1∑k=ni+1ak∑ni-1j=1aj=∑ni-1k=ni+1ak∑ni-1j=1aj |
By the way we chose the sequence (ni)∞i=0, we have 2∑ni-1k=ni+1ak>∑ni-1j=1aj and, hence,
ni-1∑k=ni+1ak∑kj=1aj≥∑ni-1k=ni+1ak∑ni-1j=1aj>12. |
Therefore,
nm∑i=1ai∑ij=1aj>nm∑i=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 |