<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] Observe that the $(2^n - 1)$th partial sum is given by $$s_{2^n - 1} = 1 + 2 \cdot \frac 1 2 + 4 \cdot \frac 1 4 + \cdots + 2^{n - 1} \cdot \frac 1 {2^{n - 1}} = \underbrace{1 + 1 + 1 + \cdots + 1}_{n \text{ summands}} = n.$$ Consequently, we have that $\lim_{n \to \infty} s_{2^n - 1} = \lim_{n \to \infty} n = \infty,$ and the series diverges.
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