<s>[INST] Finding the sum to infinity Question:
Find the sum to infinity for the following series
$$1, -\frac{1}{2}, \frac{1}{2^2}, -\frac{1}{2^3},\cdots$$
What would be the technique used to find such a sum?
 [/INST] Write the sum of the $n$ and $(n+1)^{th}$ terms as 
$$
  \frac{1}{2^n} - \frac{1}{2^{n+1}}
= \frac{1}{2^{n+1}}
$$
then sum them up to get
$$
  \sum_{n \in 2 \mathbb{N} }^\infty \frac{1}{2^{n+1}}
= \sum_{k=0}^\infty \frac{1}{2^{2k+1}}
= \frac{1}{2} \sum_{k=0}^\infty \frac{1}{4^k}
= 2/3.
$$
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