We know that the series $\sum (-1)^n/n$ converges, and clearly every other alternating harmonic series with the sign changing every two or more terms such as $$(1+\frac{1}{2}+\frac{1}{3})-(\frac{1}{4}+\frac{1}{5}+\frac{1}{6})+(\frac{1}{7}+\frac{1}{8}+\frac{1}{9})-\cdots$$ must converge. My question here is that does the series below also converge? $$\sum\frac{\textrm{sgn}(\sin(n))}{n}\quad\textrm{or}\quad\sum\frac{\sin(n)}{n|\sin(n)|}$$
Loosely speaking, the sign changes every $\pi$ terms. I'd be surprised if it doesn't converge but Wolfram Mathematica, after a couple of minutes of computing, concluded the series diverges, but I can't really trust it. My first approach (assuming the series converges) was that if we bundle up terms with the same sign like the example above every bundle must have three or four terms, and since the first three terms of all bundles make an alternating series I was going to fiddle with the remaining fourth terms but they don't make an alternating series so I guess there's no point in this approach.
from Hot Weekly Questions - Mathematics Stack Exchange
Jaeseop Ahn
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