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+1≥εn+1
and a comparison with the harmonic series shows that the series diverges.
No comments:
Post a Comment