Saturday, 14 December 2013

real analysis - Series of functions to ln



I have the following exercise:




on [1/2,1] study convergence of the derivative of i=01i(x1x)i



and show that i=0(1)i1i=ln2



I already studied convergence of the series i=01i(x1x)i and proved that it converges uniformly on every compact of [1/2,1] but I dont know the sum of the series(the sum function), also, I cant see how ln appear in the derivative


Answer



Ok I found it, the derivative of the series is equal to 1x²\sum_{i=0}^\infty (\frac{x-1}{x})^{i-1} since (\frac{x-1}{x})^{i-1} is in absolute less than 1, it converges (to x) and the whole derivative converges to \frac{1}{x}.



Since the derivative converges uniformly on the domain, we can take the integral and it should give us an equality with our series, but the integral is ln(x).. we evaluate in 1/2 and we get ln2 equals what we were looking for.



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