Prove that converges if and only if converges for any
My approach..
Assume that the latter series converges then this inequality holds for some real
for any
And thus does also the first series converge. They only differ by a constant depending on .
Now assume that diverges to for some and that converges.
This inequality holds for some
for any
But this inequality doesn't hold for any
for any
Contradiction!
My assumtition must be wrong.
Either does both diverge to or converge.
Is there something wrong? Something missing to complete the proof?
You're right. That case botherd me a bit..
But if I divided it into two cases, one case that diverge to and one diverge like your example. I.e the partial sum is oscillating between some value. Then it should be trival to show that your example diverge for any starting index.
There is really no need for cases.
Remember that series convergence is about convergence of sequences of partial sums. There is a nice results about sequences: If is a sequence and r is number then converges if and only if the sequence converges.
Realizing that is a number the result follows quickly.
This result is usually summarized by the old saying: Series convergence is completely determined by what happens to its tail.