Finding \sum_{n=1}^{\infty} (-1)^n/n - Homework Help

seanhbailey
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Homework Statement


What is \sum_{n=1}^{\infty} (-1)^n/n


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The Attempt at a Solution


I know that the alternating series \sum_{n=1}^{\infty} (-1)^{n-1}/n converges to ln(2), but I am not sure how to find this series.
 
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(-1)n-1 = (-1)n(-1)-1 = -1*(-1)n
then you can take the constant -1 outside the sum.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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