Wednesday, 23 July 2014

real analysis - Is this:suminftyn=1(1)fracn(n1)2frac1n a convergent series?



Is there someone who can show me how do I evaluate this sum :n=1(1)n(n1)21n



Note : In wolfram alpha show this result and in the same time by ratio test it's not a convince way to judg that is convergent series



Thank you for any help


Answer



The parity of n(n1)2 is 4-periodic. Thus the sequence (1)n(n1)2 equals to:

1,1,1,1,1,1,1,1,1,1,1,1,
The original series' partial sum truncated at N equals to
Kk=0(14k+114k+214k+3+14k+4)+N4K4i=1(1)i(i1)24K+4+i
where K=N41.



Then by a discussion on the partial sum we can conclude that the series is convergent.


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