Go Uniformly

Calculus Level pending

If the sequence of the function fn(x)=(1xk)n\displaystyle f_{n}(x)={(1-x^k)^n} where n,kNn,k\in\mathbb {N}, then find the value of limMlimnN=1M(m=1Nk=1m01fn(x)dxn1m2+ln(1N+1))eM.\lim_{M\to\infty}\lim_{n\to\infty}\sum_{N=1}^{M}\left(\sum_{m=1}^N\sum_{k=1}^m\sqrt[n]{\int_0^1f_n(x)dx}\frac{1}{m^2}+\ln \left(\frac{1}{N+1}\right)\right)\frac{e}{M}. Here ee is Euler's number.


Motivated by Is it 11^{\infty}? and main posting of the problem is here.

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