Calculating $\lim_{n\to \infty}{\frac{\frac{n}{1} + \frac{n-1}{2} + \frac{n-2}{3} + ... + \frac{2}{n-1} + \frac{1}{n}}{\ln(n!)}}$

Question

I have tried to use the Stolz theorem and calculate $\lim_{n\to \infty}{\frac{a_{n+1}-a_n}{b_{n+1}-b_n}}$ and i have reached $\lim_{n\to \infty}{\frac{1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n-2} + \frac{1}{n-1} + \frac{1}{n}}{\ln (n+1)}}$ but I do not know how to continue. Could someone help me? Thanks in advance.

Where is this problem from? It has been asked a couple of days ago: https://math.stackexchange.com/questions/3055274/compute-lim-n-to-infty-frac-tfracn1-tfracn-12-dots-tfrac2n-1/3055315#3055315 — A. Pongrácz, Jan 04 '19 at 12:07

score 1 · Answer 1 · answered Jan 04 '19 at 11:32

1

Hint: See the "Rate of divergence" section of the Wikipedia article about the harmonic series, or the definition of the Euler-Mascheroni constant.

answered Jan 04 '19 at 11:32

John Hughes

93,729

So i can rewrite the top part as $\ln(n) + \gamma + \frac{1}{2n}$? – Andarrkor Jan 04 '19 at 11:42
No....but you can approximate it by that (or better still, find upper and lower bounds and use the squeeze lemma). – John Hughes Jan 04 '19 at 13:26
Ok thanks I will try – Andarrkor Jan 04 '19 at 13:31

Calculating $\lim_{n\to \infty}{\frac{\frac{n}{1} + \frac{n-1}{2} + \frac{n-2}{3} + ... + \frac{2}{n-1} + \frac{1}{n}}{\ln(n!)}}$

1 Answers1