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Question Number 19053 by Tinkutara last updated on 03/Aug/17

Find the sum : \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6}

- \frac{1}{7} + \frac{1}{8} - \frac{1}{9} + \dots

Answered by ajfour last updated on 03/Aug/17

\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \frac{x^{5}}{5} - \frac{x^{6}}{6} + \dots

thus for x = 1

\ln 2 = 1 - (\frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{6} + . .)

\frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{6} + \dots = 1 - \ln 2 .

Commented by Tinkutara last updated on 03/Aug/17

Thank you very much Sir!