Multiplying McLaurin Functions & Power Series

jaejoon89
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How do you multiply McLaurin functions by "regular" power series?
For example:

If y' = sum (1, inf.) n*a_n x^n-1 and cos(x) = sum(0, inf.) (-1^n x^2n) / (2n)!, how do you find the product?

If y = sum(0, inf.) a_n x^n and sin(x) = sum(0, inf.) (-1^n x^(2n+1)) / (2n+1)!, how do you find the product of sin(x), y, and x?
 
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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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