find-the-value-lim-n-1-1-1-2-1-3-1-4-1-n-n-2-n-

Question Number 144929 by gsk2684 last updated on 30/Jun/21

find the value

\lim_{n \to \infty} {(1 + \frac{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}}{n^{2}})}^{n}

Answered by mathmax by abdo last updated on 30/Jun/21

A_{n} = {(1 + \frac{H_{n}}{n^{2}})}^{n} \Rightarrow A_{n} = e^{nlog (1 + \frac{H_{n}}{n^{2}})}

H_{n} \sim logn + γ + o (\frac{1}{n}) \Rightarrow \log (1 + \frac{H_{n}}{n^{2}}) \sim \log (1 + \frac{logn + γ + o (\frac{1}{n})}{n^{2}})

\sim \frac{\log (n) + γ + o (\frac{1}{n})}{n^{2}} \Rightarrow nlog (1 + \frac{H_{n}}{n^{2}}) \sim \frac{logn + γ + o (\frac{1}{n})}{n} \to 0 (n \to + \infty)

\Rightarrow \lim_{n \to + \infty} A_{n} = 1

Commented by gsk2684 last updated on 30/Jun/21

thank you

Commented by mathmax by abdo last updated on 30/Jun/21

you are welcome