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Question Number 40878 by prof Abdo imad last updated on 28/Jul/18

let u_0 >0 and ∀n∈N  u_(n+1) =u_n  +(1/u_n )  1) prove that (u_n )is increasing and lim u_n  =+∞  2)by consideringthe functionϕ(t)=(1/(2t+x))  prove that ∀n∈N Σ_(k=1) ^n  (1/(2k+x)) ≤(1/2)ln(1+((2n)/x))  3)find a equivalent of u_n (n→+∞)

$${let}\:{u}_{\mathrm{0}} >\mathrm{0}\:{and}\:\forall{n}\in{N} \\ $$ $${u}_{{n}+\mathrm{1}} ={u}_{{n}} \:+\frac{\mathrm{1}}{{u}_{{n}} } \\ $$ $$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left({u}_{{n}} \right){is}\:{increasing}\:{and}\:{lim}\:{u}_{{n}} \:=+\infty \\ $$ $$\left.\mathrm{2}\right){by}\:{consideringthe}\:{function}\varphi\left({t}\right)=\frac{\mathrm{1}}{\mathrm{2}{t}+{x}} \\ $$ $${prove}\:{that}\:\forall{n}\in{N}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{\mathrm{2}{k}+{x}}\:\leqslant\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\frac{\mathrm{2}{n}}{{x}}\right) \\ $$ $$\left.\mathrm{3}\right){find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \left({n}\rightarrow+\infty\right) \\ $$

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