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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 Cesarostolz theorem fails. Now the limit must be $\displaystyle \lim_{n\rightarrow \infty} \frac{f(n^2 )}{\ln(n)}=2$. cesarostolz 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=163=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 
$$\sum_{k=1}^{n^2} 1/k = 2 \ln n + \gamma + \dfrac{1}{2 n^2} + O(( \frac{1}{n} )^4) $$ 

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growth, harmonic, logarithmic, series, understand 
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