Proving $\log(2)$ with Alternating Series

[alyafey22]

It might be well-known for you that

$$\sum_{n\geq 1}\frac{(-1)^{n+1}}{n}=\log(2)$$​

There might be more than one way to prove it :)

 

[polygamma]

The typical approach is to expand $\log(1+z)$ in a Taylor series about $z=0$ and then apply Abel's theorem. But that's not particularly interesting.

 

[polygamma]

Spoiler

$$ \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{s}} = (1-2^{1-s}) \sum_{n=1}^{\infty} \frac{1}{n^{s}} = (1-2^{1-s}) \zeta(s) $$

Then

$$\lim_{s \to 1} \ \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \lim_{s \to 1 } \ (1-2^{1-s}) \zeta(s) = \lim_{s \to 1} (1-2^{1-s}) \Big( \frac{1}{s-1} + \mathcal{O}(1) \Big)$$

$$= \lim_{s \to 1} \frac{1-2^{1-s}}{s-1} = \lim_{s \to 1} \frac{2^{1-s} \log 2}{1} = \log 2 $$