ZaidAlyafey said:Prove the following
\(\displaystyle \sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360}\)
\(\displaystyle \mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2\)
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.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 .
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 .
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/.
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 .
The Harmonic sum is a mathematical series that involves adding the reciprocals of positive integers. It is denoted by Hn and can be written as 1 + 1/2 + 1/3 + ... + 1/n.
The formula for finding the Harmonic sum is Hn = 1 + 1/2 + 1/3 + ... + 1/n. This can also be written as Hn = Σi=1n 1/i, where Σ represents the summation symbol.
The Harmonic sum is important in mathematics as it helps to understand the concept of infinity and divergence. It also has applications in other fields such as physics, engineering, and computer science.
The Harmonic sum has no limit. As n approaches infinity, the Harmonic sum diverges or goes to infinity. This means that the sum of reciprocals of positive integers will continue to increase without ever reaching a finite value.
Yes, the Harmonic sum can be proven using mathematical induction. By assuming that the formula holds for a specific value of n and then proving that it holds for the next value of n, we can show that the formula holds for all positive integers. This proves the Harmonic sum.