How Do You Solve the Sum of 3^k/k! from 0 to Infinity?

[Office_Shredder]

I've been staring at this stupid problem for a while, and finally gave up and came here. The question is to evaluate the sum of 3^k/k! from 0 to infinity. Basically, I'm looking for a starting spot, since I have none. The closest thing to a strategy I've come up with is plugging it into the geometric series formula with r=3/k, but that didn't work so well

 

[AKG]

Let

$$f(x) = \sum_{k=0}^{\infty}\frac{x^k}{k!}$$

What is f'(x)? In particular, how does f'(x) relate to f(x)? What other functions relate to their derivatives in that way? If you know some stuff about differential equations, you'll know that only one type of function relates to its derivative in that way. So you can figure out what type of function f(x) is, and by plugging in x=0, you get an initial value problem which allows you to find what function f(x) is specifically. Once you know what f is, your final answer is f(3).

 

[HallsofIvy]

Actually, if you have worked with Taylor's series, you ought to be able to look at that series and recognize it immediately. If not, AKG's suggestion is excellent: take the derivative, term by term, then slap your forehead and say "oh, of course!".

 

[Office_Shredder]

After mentioning taylor's series, it's pretty obvious... unfortunately, for some obnoxious reason right before the question there was a hint about the binomial formula, so I got the impression it was supposed to be involved with that somehow. (especially since the three questions after it were).

Thanks