Infinity problem when apply infinite series representation of the modified bessel function of the second kind

Question

Gamma-Gamma distribution is a well-known distribution to characterize the optical scintillation over atmospheric turbulence, which can be found in "Laser Beam Scintillation with Applications", p89. This distribution is shown as $$f(I)=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}I^{(\alpha+\beta)/2-1}}{\Gamma(\alpha)\Gamma(\beta)}K_{\alpha-\beta} (~{2\sqrt{\alpha\beta I}}~),$$ where $I>0$. Since $f(I)$ is a probability density function(pdf), $\int_0^{\infty}f(I)dI=1$. Now I use the series expansion of $K_{\alpha-\beta}(\cdot)$ $$K_{\alpha-\beta}(2\sqrt{\alpha\beta I})=\frac{\pi}{2\sin{[\pi(\alpha-\beta)]}}\sum_{p=0}^{\infty}\Big[ \frac{\sqrt{\alpha \beta I}^{~ 2p-(\alpha-\beta)}}{\Gamma[p-(\alpha-\beta)+1]p!} - \frac{\sqrt{\alpha \beta I}^{~ 2p+(\alpha-\beta)}}{\Gamma[p+(\alpha-\beta)+1]p!} \Big]$$ in $f(I)$ and obtain the following integral $$\int_0^{\infty}f(I)dI=\frac{\pi(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma(\beta) \sin[\pi(\alpha - \beta)]} $$ $$\sum_{p=0}^{\infty}\Big[ \frac{ (\alpha\beta)^{p-\frac{\alpha-\beta}{2}} \frac{1}{p+\beta} I^{p+\beta}\mid_{0}^{\infty} }{\Gamma[p-(\alpha-\beta)+1]p!} - \frac{ (\alpha\beta)^{p+\frac{\alpha-\beta}{2}} \frac{1}{p+\alpha} I^{p+\alpha}\mid_{0}^{\infty} }{\Gamma[p+(\alpha-\beta)+1]p!} \Big]=1.$$ As we can see, the result is a infinite series and each term contains a infinitely large quantity. However, we know the integral must be 1. My question is how to explain it and how to prove the integral result in the series representation. Do you have any ideas? Thank you!