Solving the Harmonic Series and Ratio Problem

Kyousik
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Homework Statement


(infinity)sigma(n=4) [ (2 / n) - (2 / (n -1)) ]


Homework Equations


Harmonic series and ratio?


The Attempt at a Solution


It's supposed to converge to -2/3, but, I don't know how.

The first compare to a harmonic series and we see that goes to 0.

The second, we use a ratio test or something and then compare it? If we simply take the limit it goes to 0.

I don't know how to show it.
 
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Write out the first few terms of the series and see if anything occurs to you.
 
"Telescoping series"
 

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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