As part of a larger proof I'm working on (convergence in distribution of a random variable to a certain cdf), I need to show that:
$\lim_{n\to\infty} [(\frac1\pi)(\tan^{-1}(ny)+\frac\pi2)]^n=\exp\{-\frac1{\pi y}\}$
In order to do this, I have so far used the relationship $\tan^{-1}(x)+\tan^{-1}(\frac1x)=\frac\pi2$, the binomial theorem, and the series expansion of arctan to convert the above limit expression to:
$\lim_{n\to\infty} \sum_{j=0}^n {n \choose j} (-\frac1\pi)^n \left(\sum_{k=0}^\infty (\frac1{ny})^{2k+1}\frac{(-1)^k}{2k+1}\right)^n $
I feel confident that my use of the binomial theorem is on the right track, but from here I'm totally lost. I know the series expansion of the exponential that I need to reach, but I'm clueless as to how to reach it. Any help would be appreciated.