Series Convergence and Divergence

Luscinia
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Homework Statement


Determine if the following series converges or diverges. If it converges determine its sum.
Ʃ1/(i2-1) where the upper limit is n and the index i=2


Homework Equations



The General Formula for the partial sum was given:
Sn=Ʃ1/(i2-1)=3/4-1/(2n)-1/(2(n+1)

The Attempt at a Solution


I have no idea where to start. I tried to get the General Formula, but I am really confused as to how to even start. I tried to split the function with partial fractions and somehow got 1/(2(i+1))-1/(2(i-1))
 
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oops! I got it. It's a telescopic sum. I need to open my eyes a little more.
 

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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