Complex Analysis: Special Power Series

nateHI
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Homework Statement


Give an example of a power series with R=1[\itex] that converges uniformly for |z|\le 1[\itex], but such that its derived series converges nowhere for |z=1|[\itex].<br /> <br /> <h2>Homework Equations</h2><br /> R is the radius of convergence and the derived series is the term by term derivative.<br /> <br /> <h2>The Attempt at a Solution</h2><br /> I've tried a bunch of stuff but at the moment I'm leaning towards some variation of the power series for the sin function.<br /> It would be nice if there was a way to solve this without guessing and checking a bunch of times.
 
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It's OK to delete this. Posted it on accident.
 
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Sorry, I'm not sure why my latex commands aren't taking.
 
You want a forward slash to terminate, not a backslash.
 

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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