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Theorem: Let $H_n$ be the $nth$ harmonic number. Then

$$\sum_{n=1}^\infty \binom{2n}{n} H_n x^n=\frac{2}{\sqrt{1-4x}}\log\bigg(\frac{1+\sqrt{1-4x}}{2\sqrt{1-4x}} \bigg)$$

How can I simplify following hypergeometric function by using Mathematica?

Sum[Binomial[2 n, n] HarmonicNumber[n] x^n, {n, 1, Infinity}]

enter image description here

I tried FullSimplify and FunctionExpand but it doesn't work

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  • $\begingroup$ For what it's worth, -Derivative[0, 0, 1, 0][Hypergeometric2F1][1/2, 1, 1, 4*x] looks completely different from 2/Sqrt[1 - 4*x]*Log[(1 - Sqrt[1 - 4*x])/(2*Sqrt[1 - 4*x])] when I plot it. Are you sure that theorem is correct? $\endgroup$ – Pillsy Apr 11 '17 at 16:14
  • $\begingroup$ @Pillsy I think its correct. see here page 3. $\endgroup$ – vito Apr 11 '17 at 16:22
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    $\begingroup$ @Pillsy, you have a misplaced "-" in the numerator of the Log[ ] term. Plots OK after you fix that. $\endgroup$ – MikeY Apr 11 '17 at 16:31
  • $\begingroup$ @MikeY so I have. thanks! $\endgroup$ – Pillsy Apr 11 '17 at 16:35
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    $\begingroup$ Mathematica is not yet that good at simplifying parameter derivatives of hypergeometric functions, much less recognizing them to be elementary. (The general expressions are horrendous; see e.g. this.) $\endgroup$ – J. M. will be back soon Apr 11 '17 at 17:43

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