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

G iven the sequence {(U_{n})}_{n \in N} defined by U_{0} = 1 and

U_{n + 1} = f (U_{n}) where f (x) = \frac{x}{{(x + 1)}^{2}}

S how by mathematical induction that \forall n \in N^{*}

0 < U_{n} ⩽ \frac{1}{n}

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

f (x) = \frac{1}{x + 1} - \frac{1}{{(x + 1)}^{2}}

f^{'} (x) = - \frac{1}{{(x + 1)}^{2}} + \frac{2}{{(x + 1)}^{3}} = \frac{1 - x}{{(1 + x)}^{3}} ⩾ 0, \forall x \in [0, 1]

0 < U_{0} = 1 ⩽ 1 t r u e

w e a s s u m e T h a t \forall n \in N 0 < U_{n} ⩽ \frac{1}{n} ⩽ 1

s i n c e f i s i n c r e a s i n g o v e r [0, 1] \Rightarrow

\Rightarrow f (0) < f (u_{n}) ⩽ f (\frac{1}{n})

\Leftrightarrow 0 < U_{n + 1} ⩽ \frac{n}{{(n + 1)}^{2}} = \frac{n}{n + 1} . \frac{1}{n + 1} ⩽ 1 . \frac{1}{n + 1} = \frac{1}{n + 1}

\Rightarrow \forall n \in N 0 < U_{n} ⩽ \frac{1}{n}

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

Thank you ��