Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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

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

Terms of Service

Privacy Policy

Contact: info@tinkutara.com