This question is kind of an extension of a previous question I asked here.
The infinite series $$\sum\frac{\mathrm{sgn}(\sin(n))}{n}$$ does converge, but I would like to know if Dirichlet's test can be used to prove the convergence with $$b_n=\mathrm{sgn}(\sin(n)).$$ So the question is, is the series $$B_n:=\sum_{k=1}^n\mathrm{sgn}(\sin(k))$$ unbounded? Loosely speaking it is a sum of $1$'s and the sign changes every $\pi$ terms. Also it would be great to know if the series is unbounded for other (irrational) changing cycles.
from Hot Weekly Questions - Mathematics Stack Exchange
Jaeseop Ahn
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