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November 2nd, 2019, 12:16 PM  #1 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 65 Math Focus: Area of Circle  Generalization of Harmonic Series
$h(x)$ is a generalized $H_n$, and $H_n = \sum\limits_{k=1}^n \dfrac{1}{k}$. Assumptions: I started by $$\Large{f(n)= \int_1^n h(z)dz}$$ and at next page I derived at $$\Large{\int_1^n h(z) dz = \ln (n!) + (n1)k},$$ where $$\large{k=\int_0^1 h(z)dz},$$ which is going to be called EulerMascheroni constant. Now, my brain stopped working, so what should be done after $$\Large{\int_1^n h(z) dz = \ln (n!) + (n1)k}.$$ 
November 2nd, 2019, 12:32 PM  #2 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 65 Math Focus: Area of Circle 
I can just assume $\int h(z) dz = \ln (z!) + (z1)k$ and check if it holds true for $\int_1^n h(z) dz$. 
November 2nd, 2019, 07:41 PM  #3 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 65 Math Focus: Area of Circle 
How can I show $$\large{\int_0^1 h(z)dz}$$ is convergent and between 0 and 1, or it is equal to $$\large{\lim\limits_{n \rightarrow \infty} H_n  \ln (n)}$$?

November 3rd, 2019, 02:12 PM  #4  
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 65 Math Focus: Area of Circle  Quote:
I found, $$\Large{\int_{0}^1 h(z) dz = \int_0^1\left(\frac1{\ln x}+\frac{1}{1x}\right)\ dx}.$$ What to do next???  
November 3rd, 2019, 03:37 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 21,107 Thanks: 2324 
Knowing that what??

November 3rd, 2019, 05:46 PM  #6 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 65 Math Focus: Area of Circle 
Harmonic series...

November 3rd, 2019, 05:48 PM  #7 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 65 Math Focus: Area of Circle  I made a mistake... It is $$\Large{h(z) = \int_{0}^{1} \dfrac{1x^z}{1x}} dx,$$

November 4th, 2019, 02:31 AM  #8 
Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math 
$\displaystyle H_n =\ln(n) +\gamma +\epsilon_n .$ All of integrals are correct , the last one maybe has to do with the unknown Last edited by idontknow; November 4th, 2019 at 02:33 AM. 

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generalization, harmonic, series 
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