# Evaluate this infinite series

1. Oct 22, 2008

### kingwinner

1. The problem statement, all variables and given/known data
How can I compute

∑ (1 / y!) ?
y=0

2. Relevant equations
N/A

3. The attempt at a solution
In the middle of a problem from a statistics course, I got this series and forgot how to evaluate an infinite series in general and in particular this one...Please help!

2. Oct 22, 2008

### dirk_mec1

It is exp(1) or e.

3. Oct 22, 2008

### kingwinner

Why??? How did you get that?

4. Oct 22, 2008

### dirk_mec1

That's just the definition of e!

5. Oct 22, 2008

### Integral

Staff Emeritus
Definition? Maybe, but not a fundamental one.

OP
Not sure why you can't just start summing it up. What do you get? You should be able to see convergence in less then 10 terms.

6. Oct 22, 2008

### HallsofIvy

Staff Emeritus
Do you know anything about Taylor's series?

7. Oct 22, 2008

### kingwinner

Yes, I know Taylor series, but I've done it quite a while ago...

∑ (1 / y!)
y=0

∑ (1^y / y!) = e^1
y=0

Is the first series equal to the second one?
In other words, can I replace 1 by 1^y in the summand?
Here there is an ∞ involved, and I have heard that 1^∞ is an indeterminant form, so 1^∞ is not the same as 1. So after all can we still replace 1 by 1^y in the infinte series above?

8. Oct 22, 2008

### HallsofIvy

Staff Emeritus
No there is no "$\infty$" involved. That notation only means that y takes on all non-negative integer values. It is never actually equal to infinity!

9. Oct 22, 2008

### kingwinner

Um...why is it never infinity?
I have no trouble understanding that 1=1^y provided that y is finite, but when y can be infinite...how can 1=1^y=1^∞?

10. Oct 22, 2008

### HallsofIvy

Staff Emeritus
Because y has to be an integer. "infinity" is not an integer (or even a real or complex number).
$$\sum_{i= 0}^\infty a_i$$
is defined as
$$\lim_{n\rightarrow \infty} \sum_{i=0}^n a_i$$
which is a limit and also does not have i or n equal to "infinity" at any point.