<s>[INST] Is this series convergent? $1 + 1/2 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 + ...$ Is this series convergent? How to prove? 
$$1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{8} + ... + \frac{1}{8} \ (8 \times 1/8) + \frac{1}{16} + ...$$
It's equal to $\{1 + 1 + 1 + 1 + ...\}$, when I take the $(n+1)^{th}$ number and $n^{th}$ number, the ratio of them can be $\frac{1}{2}$ or $1$. I don't think this is a convergent series, but what is it? 
Thanks!
 [/INST] The most direct way to see divergence is by noticing that
$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots\leq 1+\frac{1}{2}+\frac{1}{2}+\cdots$$
where the series on the right hand side is what you have. Since the LHS diverges as the harmonic series, so must the RHS. 
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