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Question Number 29510 by abdo imad last updated on 09/Feb/18

let w_n = (1/n)( 1 +e^(1/n) +e^(2/n) + ... e^((n−1)/n) ) find lim_(n→+∞) w_n .

letwn=1n(1+e1n+e2n+...en−1n)findlimn→+∞wn.

Commented by abdo imad last updated on 11/Feb/18

we have w_n = (1/n)Σ_(k=0) ^(n−1) e^(k/n) ⇒ lim_(n→∞) w_n =lim_(n→∞) ((1−0)/n)Σ_(k=0) ^(n−1) e^((k(1−o))/n) = ∫_0 ^1 e^x dx (Reiman sum) = [e^x ]_0 ^1 =e−1 .

wehavewn=1n∑k=0n−1ekn⇒limn→∞wn=limn→∞1−0n∑k=0n−1ek(1−o)n=∫01exdx(Reimansum)=[ex]01=e−1.

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