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Question Number 98443 by Ar Brandon last updated on 14/Jun/20

Given the sequence (U_n )_(n∈N) defined by U_0 =1 and U_(n+1) =f(U_n ) where f(x)=(x/((x+1)^2 )) Show by mathematical induction that ∀n∈N^∗ 0<U_n ≤(1/n)

Giventhesequence(Un)n∈NdefinedbyU0=1andUn+1=f(Un)wheref(x)=x(x+1)2Showbymathematicalinductionthat∀n∈N∗0<Un⩽1n

Answered by maths mind last updated on 14/Jun/20

f(x)=(1/(x+1))−(1/((x+1)^2 )) f′(x)=−(1/((x+1)^2 ))+(2/((x+1)^3 ))=((1−x)/((1+x)^3 ))≥0 ,∀x∈[0,1] 0<U_0 =1≤1 true we assume That ∀n∈N 0<U_n ≤(1/n)≤1 since f is increasing over [0,1]⇒ ⇒f(0)<f(u_n )≤f((1/n)) ⇔0<U_(n+1) ≤(n/((n+1)^2 ))=(n/(n+1)).(1/(n+1))≤1.(1/(n+1))=(1/(n+1)) ⇒∀n∈N 0<U_n ≤(1/n)

f(x)=1x+1−1(x+1)2f′(x)=−1(x+1)2+2(x+1)3=1−x(1+x)3⩾0,∀x∈[0,1]0<U0=1⩽1trueweassumeThat∀n∈N0<Un⩽1n⩽1sincefisincreasingover[0,1]⇒⇒f(0)<f(un)⩽f(1n)⇔0<Un+1⩽n(n+1)2=nn+1.1n+1⩽1.1n+1=1n+1⇒∀n∈N0<Un⩽1n

Commented by Ar Brandon last updated on 14/Jun/20

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