Distribution function of Sum of IID Exponentiation Variables of Variable amount

Question

So I'm trying to determine the distribution function of a random variable, S, give:

$N \sim Geo(\frac{1}{1+\lambda}) $

$S_i \sim Exp(\mu), \forall i\in [0,N]$

$S = \Sigma^{N}_{i=0}S_i$

$S = 0$ if $N=0$

Taking the approach of first evaluating $P(S>x)$, I know that the sum of n exponentiation distributions is a Gamma distribution and by the Law of Total Probability:

$P(S>x)=\Sigma^{\infty}_{i=0}P(S>x|N=n)\times P(N=n)$

$=\Sigma^{\infty}_{n=0}[[(\frac{1}{1+\lambda})(\frac{\lambda}{1+\lambda})^n][e^{-\mu x}\Sigma^{n-1}_{r=0}\frac{(\mu x)^r}{r!}]]$ (I think)

But I'm completely stuck at that ... if it's even correct.