MHB Can the Harmonic Sum be Proven Using a Newer Method?

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    Harmonic Sum
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The discussion centers on proving the equation involving the harmonic sum, specifically $$\sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360}$$, where $$H^2_k$$ represents the square of the k-th harmonic number. The original proof was provided by D. Borwein and J. M. Borwein in 1995, but participants express a desire for a simpler, newer method of proof. Some members indicate they have not yet attempted the problem and are looking for solutions or alternative approaches. There is also a reminder about the expectations for posting solutions in the forum. The conversation highlights the complexity of the original proof and the interest in finding more accessible methods.
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Prove the following

$$\sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360}$$

$$\mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2$$​
 
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Actually , I haven't even tried it . I was reading the paper and thought it is a good exercise. I will look for a solution of my own .
 
ZaidAlyafey said:
Actually , I haven't even tried it . I was reading the paper and thought it is a good exercise. I will look for a solution of my own .
Ouch! I thought that paper was pretty brutal. I could follow the work on the integral but got lost on the Corollary work. I was kind of hoping you had a newer (and simpler) view on the matter.

-Dan
 
ZaidAlyafey said:
Actually , I haven't even tried it . I was reading the paper and thought it is a good exercise. I will look for a solution of my own .

Zaid,

When a problem is posted here in the Challenge Questions and Puzzles sub-forum, it is expected that you already have a solution ready to post after giving our members a fair amount of time to solve it, as per http://www.mathhelpboards.com/f28/guidelines-posting-answering-challenging-problem-puzzle-3875/.
 
MarkFL said:
Zaid,

When a problem is posted here in the Challenge Questions and Puzzles sub-forum, it is expected that you already have a solution ready to post after giving our members a fair amount of time to solve it, as per http://www.mathhelpboards.com/f28/guidelines-posting-answering-challenging-problem-puzzle-3875/.

I know the solution , it is in the paper . But it is very long and requires lots of things to work out . I was hoping for someone to post a newer method .

If that doesn't suit here , you can move the topic to an appropriate section .
 
ZaidAlyafey said:
I know the solution , it is in the paper . But it is very long and requires lots of things to work out . I was hoping for someone to post a newer method .

If that doesn't suit here , you can move the topic to an appropriate section .

No need, I interpreted your statement "I haven't even tried it" as meaning you had not attempted the problem. :D
 
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