Question Number 2181 by Filup last updated on 07/Nov/15
$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{series}? \\ $$$$\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{i}} \frac{\mathrm{1}}{\mathrm{2}\left({i}+\mathrm{1}\right)} \\ $$
Answered by prakash jain last updated on 08/Nov/15
$$\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{i}} \frac{\mathrm{1}}{{i}+\mathrm{1}}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}+..=\mathrm{ln}\:\mathrm{2} \\ $$
Commented by Filup last updated on 07/Nov/15
$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{simple}\:\mathrm{way}\:\mathrm{to}\:\mathrm{show}\:\mathrm{where} \\ $$$$\mathrm{this}\:\mathrm{answer}\:\mathrm{comes}\:\mathrm{from}?\:\mathrm{A}\:\mathrm{proof}? \\ $$