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Question Number 91632 by mathmax by abdo last updated on 02/May/20
$${find}\:{a}\:{equivalent}\:{of}\:{U}_{{n}} \:=\mathrm{1}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}+...+\frac{\mathrm{1}}{\sqrt{{n}}} \\ $$
Commented by mathmax by abdo last updated on 02/May/20
![we hsve U_n =Σ_(k=1) ^n (1/(√k)) the sequence n→(1/(√n)) is decreasing ⇒ U_n ∼∫_1 ^n (dt/(√t)) =[2(√t)]_1 ^n =2(√n)−2 ⇒U_n ∼ 2(√n)(n→+∞)](Q91712.png)
$${we}\:{hsve}\:{U}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{\sqrt{{k}}}\:\:\:{the}\:{sequence}\:{n}\rightarrow\frac{\mathrm{1}}{\sqrt{{n}}}\:{is}\:{decreasing}\:\Rightarrow \\ $$$${U}_{{n}} \sim\int_{\mathrm{1}} ^{{n}} \:\frac{{dt}}{\sqrt{{t}}}\:=\left[\mathrm{2}\sqrt{{t}}\right]_{\mathrm{1}} ^{{n}} \:=\mathrm{2}\sqrt{{n}}−\mathrm{2}\:\Rightarrow{U}_{{n}} \sim\:\mathrm{2}\sqrt{{n}}\left({n}\rightarrow+\infty\right) \\ $$