Tuesday, September 3, 2019

sequences and series - Convergence of suminftyn=2frac(1)n+log(1+np)sqrtnsinn



n=2(1)n+log(1+np)nsinn





A) For which pR is the series convergent?



B) For which pR is the series divergent, and what is the sum?



C) Is the series absolutely convergent for any pR?




I don't even know how to start, so far when I had a problem with parameter I could solve by using some tests. But in this case I can't see any tests I would be able to use directly.


Answer




A-B) For p0
n=2(1)n+log(1+np)nsinn=n=2(1)nnsinn+n=2log(1+np)nsinn ,
the first term converges by Leibniz criterion and the second term can be written as
n=2log(1+np)nsinn=n=2np+O(n2p)nsinn ,
which conveges for p<12 and diverges for 12p0.



For p>0 the numerator is positive so the series has only positive terms:
you obtain divergence by comparison criterion
(1)n+log(1+np)nsinnlog(1+np)1n+1Cn+1 .
C) Absolute convergence never occurs since for any p

|(1)n+log(1+np)nsinn|
has a subsequence whose sum is divergent i.e.
1+log(1+(2n)p)2nsin2n=12nsin2n+log(1+(2n)p)2nsin2n12nsin2n12n+1 .


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...