May 12th, 2019, 11:14 PM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 605 Thanks: 88  Limit with sequence
Evaluate $\displaystyle \lim_{n\rightarrow \infty} \frac{(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n})^ n }{n}$. To write it in shortterms : $\displaystyle \lim_{n\rightarrow \infty}n^{1}H_{n}^{n} \; $ , H is the harmonic series . 
May 13th, 2019, 01:03 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717 
In general $\frac{x^n}{n}\to \infty$ for $x\gt 1$. Since the harmonic series diverges, the expression $\to \infty$.


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