Question Number 31708 by gunawan last updated on 13/Mar/18
![Given sequence real function (f_n ) , f_n : [0, 2] → R ,with f_n (x)=(x^n /(1+x^n )) . n=1, 2, 3, ... a.Prove (f_n ) not uniformly convergent on [0, 2] b. Find lim_(x→∞) f_n (x) , x ∈ [0, 2]](Q31708.png)
$$\mathrm{Given}\:\mathrm{sequence}\:\mathrm{real}\:\mathrm{function}\:\left({f}_{{n}} \right)\:, \\ $$$${f}_{{n}} :\:\left[\mathrm{0},\:\mathrm{2}\right]\:\rightarrow\:\mathbb{R}\:,\mathrm{with}\: \\ $$$${f}_{{n}} \left({x}\right)=\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }\:\:.\:{n}=\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:... \\ $$$$\mathrm{a}.\mathrm{Prove}\:\left({f}_{{n}} \right)\:\mathrm{not}\:\mathrm{uniformly}\:\mathrm{convergent}\:\mathrm{on}\:\left[\mathrm{0},\:\mathrm{2}\right] \\ $$$$\mathrm{b}.\:\mathrm{Find}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{f}_{{n}} \left({x}\right)\:,\:{x}\:\in\:\left[\mathrm{0},\:\mathrm{2}\right] \\ $$