Probability- Exponential Random Variables

[Roni1985]

Homework Statement

Suppose X₁,X₂... are iid mean 1 exponential random variables. Use large deviation methodology to give a lower bound for the rate function R(a) for a>1

Homework Equations

R(a) $$\leq$$ $$\frac{-logP[Sn >n*a]}{n}$$

The Attempt at a Solution

I know that a sum of exponential random variables is Gamma (n, 1).

I'm having a problem with finding the probability.

the pdf of a gamma dist random variable is:$$\frac{x^(^n^-^1^)*e^(^-^x^)}{(n-1)!}$$
I think after transforming it, this is the pdf.

But I'm getting a very hard integral and even my TI-89 can't solve it.
I think I'm doing something wrong.

 

[Nino MMP]

I know that the rate function is the limit as n tends to infinity of R(a) so I don't know how to proceed.Any help would be greatly appreciated!