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Question Number 215739 by universe last updated on 16/Jan/25

Answered by MrGaster last updated on 19/Jan/25

$$\underset{n\to \mathrm{\infty }}{\lim }\frac{n+\underset{k=1}{\sum ^{n-1}}{n}^k}{\underset{k=1}{\sum ^n}{k}^n}$$$$=\underset{n\to \mathrm{\infty }}{\lim }\frac{n+n\left(\frac{n^n-1}{n-1}\right)}{\underset{k=1}{\sum ^n}{k}^n}$$$$=\underset{n\to \mathrm{\infty }}{\lim }\frac{n^{n+1}-1}{n^{n+1}-\underset{k=1}{\sum ^{n-1}}(n^n-k^n)}$$$$=\underset{n\to \mathrm{\infty }}{\lim }\frac{1-\frac{n}{n-1}}{1-\underset{k=1}{\sum ^{n-1}}{\left(\frac{n}{k}\right)}^n\frac{1}{n^{n+1}}}$$$$=\frac{1-0}{1-0}$$$$=1$$

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