Power Series for e^x Homework Help

islandboy401
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Homework Statement



Problem: Find the value of b for which
21.JPG



Homework Equations



Power series for (1/1+x) or in this case, power series for (1/1+b)

The Attempt at a Solution




I keep getting ln (-5/6) as the answer, but apparently the correct answer is ln (5/6). I do not see why.

Please look at my work and tell me what I am doing wrong.

21w.jpg
 
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I can't see the problem.
 
Go back and write it out with the sums, limits, and ns in their proper places and I think you will see your error.
 
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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