Finding $\lim_{n\rightarrow\infty}(1+\frac{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}{n^2})^n$

Asked Dec 24 '22 at 12:46

Active Dec 24 '22 at 13:09

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Finding $\lim_{n\rightarrow\infty}(1+\frac{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}{n^2})^n$

What I have try so for as

The limit is in the form of $1^\infty$

So Finding $\lim_{n\rightarrow\infty}(1+\frac{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}{n^2})^n=e^{\lim n\rightarrow\infty}\frac{1}{n}(1+\frac{1}{2}+\frac{1}{3}+\cdots\frac{1}{n})$

How do I solve further

Please help me

asked Dec 24 '22 at 12:46

Priti Bisht

hint: $H_n\sim \ln n$ compare it to integral since upper bound inequality is sufficient. – zwim Dec 24 '22 at 12:55
Hint: You can consider the growth of $\sum^{N}_{n=1} 1/n$ as compared to integral of $1/x$ – Rohan Didmishe Dec 24 '22 at 12:55
3

The post Find the value of $T=\mathop {\lim }\limits_{n \to \infty } {\left( {1+ \frac{{1+\frac{1}{2}+ \frac{1}{3}+ . +\frac{1}{n}}}{{{n^2}}}} \right)^n}$ is a duplicate of this one. Please find your answer(s) there. – Sarvesh Ravichandran Iyer Dec 24 '22 at 13:04

Finding $\lim_{n\rightarrow\infty}(1+\frac{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}{n^2})^n$

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