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 limnanbn exists..


Answer



To prove: n11+(1)nn diverges.




Proof: n11+(1)nn=k11+(1)2k2k+k11+(1)2k12k1=k122k+k102k1 =k11k
Because k11k diverges, n11+(1)nn diverges as well. \qed


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