let-u-k-1-1-1-2-k-n-1-1-prove-that-u-k-converges-2-let-f-x-1-1-1-2-x-n-1-with-x-0-prove-that-p-N-k-1-p-1-u-k-0-p-1-f-x-dx-k-0-p-u-k-

Tinku Tara June 4, 2023 Integration 0 Comments

Question Number 40893 by abdo.msup.com last updated on 28/Jul/18

let u_k =1−(1−(1/2^k ))^(n−1) 1)prove that Σ u_k converges 2)let f(x)=1−(1−(1/2^x ))^(n−1) with x≥0 prove that ∀p∈N Σ_(k=1) ^(p+1) u_k ≤∫_0 ^(p+1) f(x)dx≤Σ_(k=0) ^p u_k

$${let}\:{u}_{{k}} =\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{k}} }\right)^{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\Sigma\:{u}_{{k}} {converges} \\ $$$$\left.\mathrm{2}\right){let}\:{f}\left({x}\right)=\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{x}} }\right)^{{n}−\mathrm{1}} \:{with}\:{x}\geqslant\mathrm{0} \\ $$$${prove}\:{that}\:\forall{p}\in{N} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{p}+\mathrm{1}} \:{u}_{{k}} \:\leqslant\int_{\mathrm{0}} ^{{p}+\mathrm{1}} {f}\left({x}\right){dx}\leqslant\sum_{{k}=\mathrm{0}} ^{{p}} \:{u}_{{k}} \\ $$

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