Question Number 209318 by essaad last updated on 06/Jul/24
![](https://www.tinkutara.com/question/34372.png)
Answered by Frix last updated on 06/Jul/24
![We know lim_(n→∞) Σ_(k=1) ^n (1/k) =∞ k>1: (1/( (√k)))>(1/( k)) ⇒ Σ_(k=1) ^∞ (1/( (√k))) >Σ_(k=1) ^∞ (1/k) ⇒ lim_(n→∞) Σ_(k=1) ^n (1/( (√k))) =∞](https://www.tinkutara.com/question/Q209319.png)
$$\mathrm{We}\:\mathrm{know}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{{k}}\:=\infty \\ $$$${k}>\mathrm{1}:\:\frac{\mathrm{1}}{\:\sqrt{{k}}}>\frac{\mathrm{1}}{\:{k}}\:\Rightarrow\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:\sqrt{{k}}}\:>\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{k}} \\ $$$$\Rightarrow\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{\:\sqrt{{k}}}\:=\infty \\ $$
Answered by mathzup last updated on 07/Jul/24
![Σ_(k=1) ^n (1/( (√k)))∼2(√n) (n→+∞) donc la serie tend vers +∞ cad divergente.](https://www.tinkutara.com/question/Q209322.png)
$$\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{\:\sqrt{{k}}}\sim\mathrm{2}\sqrt{{n}}\:\:\:\:\:\:\left({n}\rightarrow+\infty\right)\:{donc}\:{la}\:{serie} \\ $$$${tend}\:{vers}\:+\infty\:{cad}\:{divergente}. \\ $$