Saturday, 22 July 2017

real analysis - Convergence of sumlimitsinftyn=1frac1+(1)nn



I want to check, whether n=11+(1)nn converges or diverges.



Leibniz's test failed, and ratio test just made it even more complicated, so i tried to use the comparison test, but i can't find a suitable series so that lim exists..


Answer



To prove: \sum_{n\geq 1} \frac{1+(-1)^n}{n} diverges.




Proof: \begin{align*} \sum _{n\geq 1} \frac{1+(-1)^n}{n} &= \sum _{k\geq 1} \frac{1+(-1)^{2k}}{2k} + \sum _{k\geq 1} \frac{1+(-1)^{2k-1}}{2k-1} \\ &= \sum _{k\geq 1} \frac{2}{2k} + \sum _{k\geq 1} \frac{0}{2k-1} \\\ &= \sum _{k\geq 1} \frac{1}{k} \end{align*}
Because \sum _{k\geq 1} \frac{1}{k} diverges, \sum_{n\geq 1} \frac{1+(-1)^n}{n} diverges as well. \qed


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