Proving this equation

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Translation: Because the geometric series converges uniformely for 0<a<1 in [0,a],.......
The problem is, that i can't find a theorem that allows me to switch limits in the final step. Is there such a theorem? if not can someone show me how t

>>212779
bump

>>212779
p,q>0 is also given

>>212780
Why do you even use limits here? Especially the upper limit of integral. The series wouldn't converge only for a=1 and this set has zero measure.
Also if you want to be fancy with limits and integrals you need to show that integrand satisfies Lebesgue Convergence Theorem since you are effectively pulling out the limit of series out before integral which may not always be true.

>>212798
integral limit is because the sum part of the equation is not applicable for 1 itself. as for the pulling out part... i'm pretty sure i checked that already. Maybe my idea on how to solve this is wrong?

>>212780
Yes. Uniform convergence means you can switch limits. Rudin's PMA theorem 7.11.

>>212779
Sorry OP, I have no idea. It kind of resembles a Riemann sum though, hope that might help.

What class/book is this from? I'd love to learn this sort of math.