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Question Number 212545 by MrGaster last updated on 17/Oct/24

$$\underset{n\to \mathrm{\infty }}{\mathrm{llim}}\underset{i=1}{\sum ^n}(\sqrt{1+\frac{i}{n^3}}-1)=?$$

Answered by lepuissantcedricjunior last updated on 17/Oct/24

$$\underset{\mathit{n}\to +\mathrm{\infty }}{\lim }\underset{\mathit{i}=1}{\sum ^{\mathit{n}}}(\sqrt{1+\frac{\mathit{i}}{{\mathit{n}}^3}}-1)\approx \underset{\mathit{n}\to +\mathrm{\infty }}{\lim }\underset{\mathit{i}=1}{\sum ^{\mathit{n}}}(1+\frac{\mathit{i}}{2{\mathit{n}}^3}-1)$$$$\mathit{c}\mathit{a}\mathit{r}{(1+\mathit{x})}^{\mathit{n}}=1+\mathit{n}\mathit{x}\mathit{s}\mathit{i}\mathit{x}<<<<1$$$$\approx \underset{\mathit{n}\to +\mathrm{\infty }}{\lim }\frac{1}{2{\mathit{n}}^3}\underset{\mathit{i}=1}{\sum ^{\mathit{n}}}\mathit{i}$$$$\approx \underset{\mathit{n}\to +\mathrm{\infty }}{\lim }\left(\frac{1}{2{\mathit{n}}^3}\times \frac{\mathit{n}(\mathit{n}+1)}{2}\right)$$$$\approx 0$$$$\Rightarrow \underset{\mathit{n}\to +\mathrm{\infty }}{\lim }\underset{\mathit{i}=1}{\sum ^{\mathit{n}}}(\sqrt{1+\frac{\mathit{i}}{{\mathit{n}}^3}}-1)=0$$

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