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let-u-n-k-1-n-1-k-n-2-1-verify-that-x-x-2-2-ln-1-x-x-2-prove-that-u-n-is-convergente-and-find-its-limit-

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Question Number 30173 by abdo imad last updated on 18/Feb/18
let u_n = Π_(k=1) ^n (1+(k/n^2 )) 1. verify that x−(x^2 /2) ≤ln(1+x)≤x 2. prove that (u_n ) is convergente and find its limit.
letun=∏k=1n(1+kn2)1.verifythatx−x22⩽ln(1+x)⩽x2.provethat(un)isconvergenteandfinditslimit.
Commented by abdo imad last updated on 21/Feb/18
let put ϕ(t)= ln(1+x)−(x−(x^2 /2)) for x≥0 we have ϕ^′ (t)= (1/(1+x)) −(1−x)=((1−(1−x^2 ))/(1+x))= (x^2 /(1+x)) ≥0 so ϕ is increasing on [0,+∞[ we have ϕ(0)=0 ≥0 ⇒ ∀ x≥0 ϕ(x)≥0 we use the same method for the function ψ(x)= x−ln(1+x) and we show that ψ(x)≥0 2) we have ln(u_n )= Σ_(k=1) ^n ln(1+ (k/n^2 )) but (k/n^2 ) −(k^2 /(2n^4 )) ≤ln (1+(k/n^2 )) ≤(k/n^2 ) ⇒ Σ_(k=1) ^n (k/n^2 )− Σ_(k=1) ^n (k^2 /(2n^4 )) ≤ Σ_(k=1) ^n ln(1+(k/n^2 ))≤ Σ_(k=1) ^n (k/n^2 ) ⇒ (1/(2n^2 ))n(n+1) −(1/(2n^4 ))((n(n+1)(2n+1))/6)≤ln(u_n )≤ (1/(2n^2 ))n(n+1)⇒ ((n+1)/(2n)) −(1/(12n^3 ))(n+1)(2n+1) ≤ln(u_n )≤ ((n+1)/(2n)) but lim_(n→∞) ((n+1)/(2n)) −(((n+1)(2n+1))/(12n^3 ))= (1/2) and lim_(n→∞) ((n+1)/(2n))=(1/2) ⇒ lim_(n→∞) ln(u_n )=(1/2) ⇒ lim_(n→∞) u_n = (√e) .
letputφ(t)=ln(1+x)−(x−x22)forx⩾0wehaveφ′(t)=11+x−(1−x)=1−(1−x2)1+x=x21+x⩾0soφisincreasingon[0,+∞[wehaveφ(0)=0⩾0⇒∀x⩾0φ(x)⩾0weusethesamemethodforthefunctionψ(x)=x−ln(1+x)andweshowthatψ(x)⩾02)wehaveln(un)=∑k=1nln(1+kn2)butkn2−k22n4⩽ln(1+kn2)⩽kn2⇒∑k=1nkn2−∑k=1nk22n4⩽∑k=1nln(1+kn2)⩽∑k=1nkn2⇒12n2n(n+1)−12n4n(n+1)(2n+1)6⩽ln(un)⩽12n2n(n+1)⇒n+12n−112n3(n+1)(2n+1)⩽ln(un)⩽n+12nbutlimn→∞n+12n−(n+1)(2n+1)12n3=12andlimn→∞n+12n=12⇒limn→∞ln(un)=12⇒limn→∞un=e.

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