Dec 26, 2013 Thread starter #1 ZaidAlyafey Well-known member MHB Math Helper Jan 17, 2013 1,667 It might be well-known for you that \(\displaystyle \sum_{n\geq 1}\frac{(-1)^{n+1}}{n}=\log(2)\) There might be more than one way to prove it
It might be well-known for you that \(\displaystyle \sum_{n\geq 1}\frac{(-1)^{n+1}}{n}=\log(2)\) There might be more than one way to prove it
Dec 26, 2013 #2 Random Variable Well-known member MHB Math Helper Jan 31, 2012 253 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.
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.
Dec 26, 2013 #3 Random Variable Well-known member MHB Math Helper Jan 31, 2012 253 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 $$\lim_{s \to 1} \ \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \lim_{s \to 1 } \ (1-2^{1-s}) \zeta(s) = \lim_{s \to 1} (1-2^{1-s}) \Big( \frac{1}{s-1} + \mathcal{O}(1) \Big)$$ $$= \lim_{s \to 1} \frac{1-2^{1-s}}{s-1} = \lim_{s \to 1} \frac{2^{1-s} \log 2}{1} = \log 2 $$ Last edited: Dec 26, 2013
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 $$\lim_{s \to 1} \ \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \lim_{s \to 1 } \ (1-2^{1-s}) \zeta(s) = \lim_{s \to 1} (1-2^{1-s}) \Big( \frac{1}{s-1} + \mathcal{O}(1) \Big)$$ $$= \lim_{s \to 1} \frac{1-2^{1-s}}{s-1} = \lim_{s \to 1} \frac{2^{1-s} \log 2}{1} = \log 2 $$