Show that $\sum{\frac{x^n}{(1+x^n)^n}}$ converges uniformly

real-analysis sequences-and-series

Show that $$\sum{\frac{x^n}{(1+x^n)^n}}$$ converges uniformly on $[0,1].$

I am sorry but for this exercise I got exactly nothing. It seems to be difficult.


Solution 1:

This problem has already been asked in the American Mathematical Monthly (number of the problem 10840 in the volume 107, number 7, December 2000, page 950) and a solution can be found in the volume 109, number 4 (April 2002), pages 398-399 of the American Mathematical Monthly.

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