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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-




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]
$$\mathrm{Given}\:\mathrm{sequence}\:\mathrm{real}\:\mathrm{function}\:\left({f}_{{n}} \right)\:, \\ $$$${f}_{{n}} :\:\left[\mathrm{0},\:\mathrm{2}\right]\:\rightarrow\:\mathbb{R}\:,\mathrm{with}\: \\ $$$${f}_{{n}} \left({x}\right)=\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }\:\:.\:{n}=\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:… \\ $$$$\mathrm{a}.\mathrm{Prove}\:\left({f}_{{n}} \right)\:\mathrm{not}\:\mathrm{uniformly}\:\mathrm{convergent}\:\mathrm{on}\:\left[\mathrm{0},\:\mathrm{2}\right] \\ $$$$\mathrm{b}.\:\mathrm{Find}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{f}_{{n}} \left({x}\right)\:,\:{x}\:\in\:\left[\mathrm{0},\:\mathrm{2}\right] \\ $$

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