After analysing the outage probability of a single relay selection system, I got to the following form:
$P = 1 + \sum\limits_{k = 1}^K {\left( \begin{array}{l}
K\\
k
\end{array} \right){{\left( { - 1} \right)}^k}2\sqrt {\frac{{k{\gamma _o}{c_p}}}{{{\lambda _{SR}}{\lambda _{RD}}}}} {e^{ - \frac{{k{\gamma _o}}}{{{\lambda _{SR}}}}}}{K_1}\left( {2\sqrt {\frac{{k{\gamma _o}{c_p}}}{{{\lambda _{SR}}{\lambda _{RD}}}}} } \right)}$. When $\lambda_{SR}$ and $\lambda_{RD}$ go to infinity we can use the approximation: ${K_1}\left( x \right) \sim \frac{1}{x}$ (where ${K_1}\left( x \right)$ is the first-ordered modified Bessel function of the second kind) to get an asymptotic of the above formula to be:
$P = 1 + \sum\limits_{k = 1}^K {\left( \begin{array}{l}
K\\
k
\end{array} \right){{\left( { - 1} \right)}^k}{e^{ - \frac{{k{\gamma _o}}}{{{\lambda _{SR}}}}}}}$. However, the original and the approximation forms are not closed when $\lambda_{SR}$ and $\lambda_{RD}$ go to infinity. Here is the test figure:
Can somebody give me some hint how these form behave like that? Thank you very much. Best Regards, Binh.
Your asymptotic expansion for the Bessel function $K_1$ is wrong (from which source is it?) and therefore the resulting formula for $P$ too. The correct one is (derived from e.g. Abramowitz and Stegun 9.7.2 or http://dlmf.nist.gov/10.40#i): $$K_1(x) = \sqrt{\frac{\pi}{2x}}e^{-x}\left(1 + \frac{3}{8x}+ O(x^{-2})\right)$$