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I was learning about the harmonic series back in college to which the professor said "There is no known closed form for the harmonic sum", I felt that was strange given that the sum didn't look complicated at all. So, I took a pen and paper and started exploring a way to find a closed form. In the end I found this function :

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

$M = \phi - ln(2\phi+1)$

When evaluated for at $f(n+1)$ I get a really close approximation to $H_n$. What got me even excited is that the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ popped up naturally when I was deriving this equation. Now am wondering, could we use this function as an approximation to the Digamma function given that $ψ(n) = H_{n-1}+\gamma$?

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  • $\begingroup$ How did you find $f(x)$? Maybe by calculating $\int_1^x\frac{\sqrt{4t^2+1}+1}{2t^2}dt$? $\endgroup$
    – Тyma Gaidash
    Commented Mar 2 at 18:47
  • $\begingroup$ Yes I integrated that formula $\endgroup$
    – Motez Ouledissa
    Commented Mar 2 at 18:52

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