prove $\sum\limits_{n\geq 1} (-1)^{n+1}\frac{H_{\lfloor n/2\rfloor}}{n^3} = \zeta^2(2)/2-\frac{7}{4}\zeta(3)\log(2)$

Prove the following

$$\sum\limits_{n\geq 1}(-1)^{n+1}\frac{H_{\lfloor n/2\rfloor}}{n^3} = \frac{1}{2}\zeta(2)^2-\frac{7}{4}\zeta(3)\log(2)$$

I was able to prove the formula above and interested in what approach you would take .

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Then you just want other proofs? You should state that. Else people might simply repeat what you just know. Also, people tend to find "Prove $X$. The end." questions a little rude. –  Pedro Tamaroff Aug 14 '13 at 0:08
(As you can see, someone has downvoted.) –  Pedro Tamaroff Aug 14 '13 at 0:10
Sorry but I do not think "I was able to prove the formula above and interested in what approach you would take" is accurate. Since we are not guinea pigs on whom you perform experiments (or are we?), what should interest you are proofs different from yours. Hence the post should explain the proof you found. –  Did Aug 14 '13 at 0:42
@Did, I never said that . I registered in many forums and there is a sub-forum called , challenges , where you post difficult questions. If this is not allowed here , then my apology I will delete the challenge. –  Zaid Alyafeai Aug 14 '13 at 0:54
@Did You are being unduly hostile. While the OP may have been a little terse, and a little lacking in details, challenge problems are a time-honored tradition of math forums. Now, it may not be the etiquette of math.se to partake in open problem challenges, but I assure you it is absolutely commonplace on most other forms. Experimenting? Guinea pigs? A little dramatic. –  Alex Youcis Aug 14 '13 at 0:59