<s>[INST] Prove that $\sum_{k=0}^{\infty} (k-1)/2^k = 0$ How to prove that this series converges, and that the limit is 0 ?
 [/INST] $$
\sum_{k=0}^\infty \frac{k-1}{2^k} = -1 + \frac{1}{2}\sum_{k=1}^\infty \frac{k-1}{2^{k-1}}.
$$
If we call $n=k-1$, then
$$
\sum_{k=0}^\infty \frac{k-1}{2^k} = -1 + \frac{1}{2}\sum_{n=0}^\infty \frac{n}{2^n} = -1 + \frac{1}{2}\sum_{n=0}^\infty \frac{1}{2^n}+ \frac{1}{2}\sum_{n=0}^\infty \frac{{n-1}}{2^n}.
$$
Therefore
$$
\frac{1}{2}\sum_{k=0}^\infty \frac{{k-1}}{2^k} = -1 + \frac{1}{2}\sum_{n=0}^\infty \frac{1}{2^n} = 0.
$$
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