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#### cristianoceli

##### New member

- Jun 16, 2016

- 12

a) Prove that series $\displaystyle\sum_{n=1}^\infty{\displaystyle\frac{ln(1+nx)}{nx^n}}$

converges uniformly on the set $ S = [2, \infty) $.

b Prove that series $\displaystyle\sum_{n=1}^\infty{(-1)^{n+1} \displaystyle\frac{e^{-nt}}{\sqrt[ ]{n+t^2}}}$

converges uniformly on the set $ S = [0, \infty) $.

My attempt:

a) The fuctions $f_n(x)=\displaystyle\frac{ln(1+nx)}{nx^n}$ are decreasing (I dont how prove). Therefore $|f_n(x)|\leq f_n(2)$

b) I do not know how to start

Thanks