N=0 vs N=1 in Homework Equations: What's the Difference?

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Homework Statement


taylorseries.png



Homework Equations





The Attempt at a Solution



I have the same final answer except without the X/2 + power series(n=1), I just have the same power series except its n=0.

Is there any reason why they are pulling out n=0 and changing it to n=1?

I did a similar problem which seems I could do the same thing but they left it as n=0.
 
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The product in the numerator, 1x3x5x...x(2n-1), doesn't really work out for n=0.
 

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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