# Moment Generating Functions

#### silver

##### New Member
The discrete random variable X has probability function
p(x)=4/(5^X+1) X=0, 1, 2,...

Derive the MGF of X and use it to find E(X) and V(X).

I have managed to get this far:

Mx(t) = Σ(e^tX)(4/(5^X+1))

e^(tX) = 1 + tX + (t^2/2!)X^2 + (t^3/3!)X^3 + ...

So Mx(t) = Σ(4/(5^X+1)) + tΣ(4/(5^X+1)) + (t^2/2!)Σ(4/(5^X+1)) + (t^3/3!)Σ(4/(5^X+1)) + ...

But I have no idea where to go from here, any help would be much appreciated!

#### BGM

##### TS Contributor
First of all do you mean the p.m.f. is in the form of

$$\Pr\{X = x\} = \frac {c} {5^x + 1}, x = 0, 1, 2, \ldots$$

Then from wolframalpha,

http://www.wolframalpha.com/input/?i=sum+1/(5^x+++1),+x+=+0+to+inf

the normalizing constant is not equal to 4, and the series itself does not has a nice closed form solution (just in terms of the digamma function).

Is there anything wrong?