# Capital Recovery Factor for Hyperbolic Discounting

I'm looking for a closed-form capital recovery factor when hyperbolic discounting is used. The Wikipedia article on hyperbolic discounting has this formula for the "present value of a series of equal annual cash flows in arrears discounted hyperbolically": $$V = P \frac{\ln(1 + kd)}{k}\ ,$$ but does not include any references.

If I attempt to sum the series $$\sum_{t=1}^n 1/(1 + rt)$$ however, WolframAlpha gives me a more complicated formula involving the digamma function.

I would appreciate a reference for this CRF, if one exists. A derivation would be a bonus. Thanks!

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• It looks like they've just approximated the sum with an integral – B.Martin yesterday
• The sum will not have an exact form. en.m.wikipedia.org/wiki/Harmonic_number. This pages gives some nice asymptotic expansions in the case $r=1$. The logarithm approximation is pretty good though. – B.Martin yesterday