Showing uniform convergence (or lack of) on [0,1]

dracond
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Homework Statement



Hello! I've been tasked with figuring out if the following is uniformly convergent on [0,1] and I could use push in the right direction:

sin(nx)


Homework Equations





The Attempt at a Solution



Would picking M=1 in the weierstrass-m test show it not uniformly convergent?
 
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Does it even converge pointwise?
 
Oops, it oscillates between [-1,1], no?
so does not converge pointwise.
 

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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