 My Math Forum Cannot understand harmonic series logarithmic growth

 Real Analysis Real Analysis Math Forum

 October 16th, 2019, 06:15 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math Cannot understand harmonic series logarithmic growth I know that $\displaystyle 1+1/2+1/3+...+1/n =\ln(n)+\gamma+\epsilon_{k}=f(n)$. where $\displaystyle \epsilon_{k}\approx 1/2n$. The problem is this sum : $\displaystyle 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n^2 }$=? , can we go like $\displaystyle H_{n^2 } =f(n^2 )$? If yes then (*) $\displaystyle \: \displaystyle 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n^2 }=\ln(n^2 )+\gamma +\epsilon(n^2 )$. Im trying to solve the limit $\displaystyle \displaystyle l=\lim_{n\rightarrow \infty } \frac{\sum_{i=1}^{n^2 }i^{-1}}{\ln(n)}$ using (*) , since Cesaro-stolz theorem fails. Now the limit must be $\displaystyle \lim_{n\rightarrow \infty} \frac{f(n^2 )}{\ln(n)}=2$. cesaro-stolz theorem cannot fail if we know the number of elements of the difference (which is a sum too) $\displaystyle d_n =H_{n^2 +2n+1} -H_{n}$. For example , n=3 , number of elements=16-3=13 ; n=2 , number of elements=7...etc . Can we express $\displaystyle d_n$ knowing that the number of elements is $\displaystyle n^2 +2n +1 -n=n^2 +n +1$? Last edited by idontknow; October 16th, 2019 at 06:48 AM. October 16th, 2019, 12:11 PM #2 Senior Member   Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 64 Math Focus: Area of Circle $$\sum_{k=1}^{n^2} 1/k = 2 \ln n + \gamma + \dfrac{1}{2 n^2} + O(( \frac{1}{n} )^4)$$ Thanks from idontknow Tags growth, harmonic, logarithmic, series, understand Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post hydronate Math 7 June 22nd, 2018 02:34 PM William Labbett Number Theory 0 October 11th, 2014 04:57 AM Daltohn Calculus 3 March 2nd, 2014 01:31 PM tolko Real Analysis 1 February 11th, 2012 03:28 PM brunojo Real Analysis 11 December 2nd, 2007 08:49 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      