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lim-k-i-1-k-1-i-




Question Number 185328 by liuxinnan last updated on 20/Jan/23
lim_(k→+∞) Σ_(i=1) ^k (1/( (√i)))=
$${lim}_{{k}\rightarrow+\infty} \underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{\mathrm{1}}{\:\sqrt{{i}}}= \\ $$
Answered by aleks041103 last updated on 20/Jan/23
since (√i)≤i⇒(1/( (√i)))≥(1/i)  ⇒Σ_(i=1) ^k (1/( (√i)))>Σ_(i=1) ^k (1/i)  since Σ_(i=1) ^k (1/i) diverges, then  lim_(k→∞)  Σ_(i=1) ^k (1/( (√i))) →+∞
$${since}\:\sqrt{{i}}\leqslant{i}\Rightarrow\frac{\mathrm{1}}{\:\sqrt{{i}}}\geqslant\frac{\mathrm{1}}{{i}} \\ $$$$\Rightarrow\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{\mathrm{1}}{\:\sqrt{{i}}}>\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{\mathrm{1}}{{i}} \\ $$$${since}\:\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{\mathrm{1}}{{i}}\:{diverges},\:{then} \\ $$$$\underset{{k}\rightarrow\infty} {{lim}}\:\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{\mathrm{1}}{\:\sqrt{{i}}}\:\rightarrow+\infty \\ $$

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