How do I proceed with this series

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I am trying to prepare for the final and am not sure how to proceed with this. Am I just supposed to plug in and cancel terms?

--> determine whether diver/conv. If conv, find the sum:

summation from 1 to infinity of 15(5/6)^(n-1)
 
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hmm

Seems like this can also be seen as 15 * 1/(5/6) * (5/6)^n which reminds me of a geometric series...
 
thank you.
 
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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