Is there someone who can show me how do I evaluate this sum :∞∑n=1(−1)n(n−1)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(n−1)2 is 4-periodic. Thus the sequence (−1)n(n−1)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
K∑k=0(14k+1−14k+2−14k+3+14k+4)+N−4K−4∑i=1(−1)i(i−1)24K+4+i
where K=⌊N4⌋−1.
Then by a discussion on the partial sum we can conclude that the series is convergent.
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