Thursday, 2 January 2020

sequences and series - Convergence of the following sum



Does the following sum converge? n=1sin2(n)n
I tried the ratio test and got that ρ=0 which means that the series converges absolutely. However, Mathematica and Wolfram Alpha do not give a result when trying to find its convergence. Am I wrong?


Answer



Yes, you are wrong. The ratio test is inconclusive, and the series diverges.



Note that there is some ε>0 such that sin2(n)+sin2(n+1)>ε for all n. This is because if n is close to a multiple of π, n+1 will not be. Thus sin2(n)n+sin2(n+1)n+1εn+1

and a comparison with the harmonic series shows that the series diverges.


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