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


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

$l e t u_{k} = 1 - {(1 - \frac{1}{2^{k}})}^{n - 1}$ $1) p r o v e t h a t Σ u_{k} c o n v e r g e s$ $2) l e t f (x) = 1 - {(1 - \frac{1}{2^{x}})}^{n - 1} w i t h x ⩾ 0$ $p r o v e t h a t \forall p \in N$ $\sum_{k = 1}^{p + 1} u_{k} ⩽ \int_{0}^{p + 1} f (x) d x ⩽ \sum_{k = 0}^{p} u_{k}$
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