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Question Number 119032 by Algoritm last updated on 21/Oct/20

Answered by 1549442205PVT last updated on 22/Oct/20

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...-\frac{1}{2\mathrm{n}}=\frac{1}{\mathrm{n}+1}+...+\frac{1}{2\mathrm{n}}$$$$\mathrm{Put}\mathrm{S}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...-\frac{1}{2\mathrm{n}}$$$$\Rightarrow \mathrm{S}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2\mathrm{n}-1}+\frac{1}{2\mathrm{n}}$$$$-2(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2\mathrm{n}})$$$$=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2\mathrm{n}-1}+\frac{1}{2\mathrm{n}}$$$$-(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{\mathrm{n}})=\frac{1}{\mathrm{n}+1}+...+\frac{1}{2\mathrm{n}}$$$$(\mathbf{Q}.\mathbf{E}.\mathbf{D})$$

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