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Question Number 145444 by mathdanisur last updated on 04/Jul/21

$$\frac{1}{2}-\frac{3}{4}+\frac{7}{8}-\frac{15}{16}+...=?$$

Answered by Olaf_Thorendsen last updated on 05/Jul/21

$$\mathrm{S}=\underset{n=0}{\sum ^{\mathrm{\infty }}}{(-1)}^n\frac{2^{n+1}-1}{2^{n+1}}$$$$\mathrm{S}=\underset{n=0}{\sum ^{\mathrm{\infty }}}{(-1)}^n(1-\frac{1}{2^{n+1}})$$$$\mathrm{S}=\underset{n=0}{\sum ^{\mathrm{\infty }}}{(-1)}^n+\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n}{2^n}$$$$\underset{n=0}{\sum ^{\mathrm{\infty }}}{(-1)}^n\mathrm{diverges}$$$$\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n}{2^n}=-1+\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n}{2^n}$$$$=-1+\frac{1}{1+\frac{1}{2}}=-\frac{1}{3}\mathrm{converges}$$$$\mathrm{S}\mathrm{diverges}$$

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