F(x) = \(\frac{1}{B(p,q)}\) \(\frac{X^{p-1}}{(1+X)^{p+q}}\)

Harmonic mean = \(\frac{\displaystyle\sum_{n=0}^{\infty} F(X_n)}{\displaystyle\sum_{n=0}^{\infty} \frac{F(X_n)}{X}}\)

and we know sum change into integration in continuous series.

Harmonic mean of F(x) = \(\frac{\frac{1}{B(p,q)}\int_0^\infty \frac{X^{p-1}}{(1+X)^{p+q}}}{\frac{1}{B(p,q)}\int_0^\infty \frac{X^{p-1}}{X(1+X)^{p+q}}}\)

= \(\frac{\int_0^\infty\frac{X^{p-1}}{(1+X)^{p+q}}}{\int_0^\infty\frac{X^{(p-1)-1}}{(1+X)^{p+q+1}}}\) = \(\frac{B(p,q)}{B(p-1,q+1)}\) \(\Big(\because \int_0^\infty \frac{x^{m-1}}{(1+x)^{m+n}} = B(m,n)\Big)\)

= \(\frac{p!q!}{(p+q)!}\) x \(\frac{(p+q)!}{(p-1)!(q+1)!}\)

= \(\frac{p(p-1)!q!}{(p-1)!(q+1)q!}\) = \(\frac{p}{q+1}\)