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let-x-n-1-1-n-x-with-x-gt-1-1-calculate-lim-x-1-x-and-lim-x-x-2-prove-that-x-1-2-x-o-2-x-x-3-prove-that-is-decreasing-and-convexe-fuc

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Question Number 62425 by mathmax by abdo last updated on 21/Jun/19
let ξ(x) =Σ_(n=1) ^∞ (1/n^x ) with x>1 1) calculate lim_(x→1^+ ) ξ(x) and lim_(x→+∞) ξ(x) 2) prove that ξ(x) =1+2^(−x) +o(2^(−x) ) (x→+∞) 3) prove that ξ is decreasing and convexe fucntion on]1,+∞[
letξ(x)=∑n=1∞1nxwithx>11)calculatelimx→1+ξ(x)andlimx→+∞ξ(x)2)provethatξ(x)=1+2−x+o(2−x)(x→+∞)3)provethatξisdecreasingandconvexefucntionon]1,+∞[

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