<s>[INST] Is $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}-\frac{1}{8}-\dots$ convergent? 
Let  $a_n= (-1)^k$ ,when $2^k \leq n < 2^{k+1} $.
 Then determine if $\sum_{n=1}^{\infty}\frac{a_n}{n}$ converge or not.

[my attempt]
I try to apply Dirichlet tset but I fail.
So I tried following method
$$1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}-\frac{1}{8}-\dots \geq  1-\left(\frac{1}{2}+\frac{1}{2}\right)+\left(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right)-\frac{1}{8}-\dots = 1 - 1 +1 + \dots $$ However this has no information so I fail,too
May I ask you how to solve this problem?
 [/INST] Bunching terms with the same sign together already gives you enough information. Each such bunched term has a magnitude of at least $\frac12$, which means that the series diverges by Cauchy's test (all sums of sufficiently far sequences of arbitrary length must be less than any $\varepsilon$ for the series to converge).
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