MHB What is the Interesting Euler Sum Proven by this Equation?

AI Thread Summary
The discussion centers on proving the Euler sum involving harmonic numbers and logarithmic functions. The equation presented is linked to the series of harmonic numbers, specifically the relationship between \( H_{2k-1} \) and the sum of reciprocals of odd integers. Participants clarify the distinction between \( H_{2k} \) and \( H_{2k-1} \), with one contributor admitting an earlier mistake in their calculations. The conversation also explores the derivation of the series involving harmonic numbers and their contributions to the overall sum. The thread emphasizes the complexity of evaluating these sums accurately.
alyafey22
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Prove the following Euler sum

$$\sum_{k\geq 1}\left(1+\frac{1}{3}+\cdots +\frac{1}{2k-1} \right) \frac{x^{2k}}{k}=\frac{1}{4}\ln^2\left( \frac{1+x}{1-x}\right)$$​
 
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I get that

$$\sum_{k=1}^{\infty} H_{2k-1} \frac{x^{2k}}{k} = \frac{1}{2} \Big( \ln^{2}(1-x) + \ln^{2}(1+x) \Big)$$
The generating function for the harmonic numbers is $ \displaystyle -\frac{\ln(1-x)}{1-x} $.

So

$$ \sum_{k=1}^{\infty} (-1)^{k} H_{k} x^{k} = - \frac{\ln(1+x)}{1+x}$$

And

$$ \sum_{k=1}^{\infty} H_{k} x^{k} - \sum_{k=1}^{\infty} (-1)^{k} H_{k} x^{k} = 2 \sum_{k=1}^{\infty} H_{2k-1} x^{2k-1} = -\frac{\ln(1-x)}{1-x} + \frac{\ln(1+x)}{1+x}$$

Now integrate.

$$

\sum_{k=1}^{\infty} H_{2k-1} \frac{x^{2k}}{k} = - \int \frac{\ln(1-x)}{1-x} \ dx + \int \frac{\ln(1+x)}{1+x} \ dx $$

$$ = \frac{1}{2} \Big( \ln^{2}(1-x) + \ln^{2}(1+x) \Big) + C$$

And when $x$ is zero, $ \displaystyle \frac{1}{2} \Big( \ln^{2}(1-x) + \ln^{2}(1+x) \Big) =0$. So $C = 0 $.
 
I think you should consider the harmonic numbers

$$H_{2k}+ H'_{2k}$$ , that's way you got a different answer .
 
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But our answers aren't equivalent.
 
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$$1+\frac{1}{3}+\frac{1}{5}+\cdots +\frac{1}{2k-1}= \left(1+\frac{1}{2}+\frac{1}{3} +\cdots +\frac{1}{2k-1}+\frac{1}{2k}\right)-\frac{1}{2}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{k} \right)=H_{2k} -\frac{1}{2}H_k$$

- - - Updated - - -

I think you are evaluating $$H_{2k-1}\neq 1+\frac{1}{3}+\frac{1}{5}+\cdots +\frac{1}{2k-1}$$
 
Yeah. My mistake. Sorry.
 
Redo (Smile)
$ \displaystyle \sum_{k =1}^{\infty} \left(1+\frac{1}{3}+\cdots +\frac{1}{2k-1} \right) \frac{x^{2k}}{k} = \sum_{k=1}^{\infty} \left( H_{2k} - \frac{1}{2} H_{k} \right) \frac{x^{2k}}{k} $$\displaystyle \sum_{k=1}^{\infty} H_{k} x^{k} + \sum_{k=1}^{\infty} (-1)^{k} H_{k} x^{k} = 2 \sum_{k=1}^{\infty} H_{2k} x^{2k} = - \frac{\ln(1-x)}{1-x} - \frac{\ln(1+x)}{1+x}$$\displaystyle \sum_{k=1}^{\infty} H_{2k} \frac{x^{2k}}{k} = - \frac{1}{2} \int \frac{\ln(1-x)}{x(1-x)} \ dx - \frac{1}{2} \int \frac{\ln(1+x)}{x(1+x)} \ dx $

$ \displaystyle = \text{Li}_{2} (x) + \frac{1}{2} \ln^{2}(1-x) + \text{Li}_{2}(-x) + \frac{1}{2} \ln^{2}(1+x) + C $ (partial fractions)

$ \displaystyle = \frac{1}{2} \text{Li}_{2}(x^{2}) + \frac{1}{2} \ln^{2}(1-x) + \frac{1}{2} \ln^{2}(1+x) + C$The constant of integration must be zero.$ \displaystyle \sum_{k=1} H_{k} \frac{x^{2k}}{k} = \text{Li}_{2}(x^{2})+ \frac{1}{2} \ln^{2}(1-x^{2})$So

$ \displaystyle \sum_{k=1}^{\infty} \left( H_{2k} - \frac{1}{2} H_{k} \right) \frac{x^{2k}}{k}= \frac{1}{2} \text{Li}_{2}(x^{2}) + \frac{1}{2} \ln^{2}(1-x) + \frac{1}{2} \ln^{2}(1+x) - \frac{1}{2} \text{Li}_{2}(x^{2}) - \frac{1}{4} \ln^{2}(1-x^{2}) $

$ \displaystyle = \frac{1}{2} \ln^{2}(1-x) + \frac{1}{2} \ln^{2}(1+x) - \frac{1}{4} \Big( \ln(1-x) + \ln(1+x) \Big)^{2} $

$ \displaystyle = \frac{1}{4} \ln^{2}(1-x) + \frac{1}{4} \ln^{2}(1+x) - \frac{1}{2} \ln(1-x) \ln(1+x)$

$ \displaystyle = \Big( \frac{1}{2} \ln(1-x) - \frac{1}{2} \ln(1+x) \Big)^{2} $

$ \displaystyle = \frac{1}{4} \ln^{2} \Big( \frac{1+x}{1-x} \Big) $
 
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Random Variable said:
$\displaystyle \sum_{k=1}^{\infty} H_{k} x^{k} + \sum_{k=1}^{\infty} (-1)^{k} H_{k} x^{k} = 2 \sum_{k=1}^{\infty} H_{2k} x^{2k} = \cdots $

I am not convinced of this step , how did you get $$H_{2k}$$ ?
 
$ \displaystyle \sum_{k=1}^{\infty} H_{k} x^{k} + \sum_{k=1}^{\infty} (-1)^{k} H_{k} x^{k} = (H_{1} x + H_{2} x^{2} + H_{3}x^{3}+ H_{4}x^{4} \ldots) + (-H_{1} x^{1} + H_{2} x^{2} - H_{3}x^{3} + H_{4}x^{4} \ldots ) $

$ \displaystyle = 2 H_{2}x^{2} + 2 H_{4}x^{4} + \ldots $

$ \displaystyle = 2 \sum_{k=1}^{\infty} H_{2k} x^{2k} $
 

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