How Can I Evaluate the Infinite Series (1/y!) for y = 0 to Infinity?

[kingwinner]

Homework Statement

How can I compute
∞
∑ (1 / y!) ?
y=0

Homework Equations

N/A

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!

 

[dirk_mec1]

It is exp(1) or e.

 

[kingwinner]

Why? How did you get that?

 

[dirk_mec1]

That's just the definition of e!

 

[Integral]

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.

 

[HallsofIvy]

Do you know anything about Taylor's series?

 

[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 infinite series above?

 

[HallsofIvy]

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!

 

[kingwinner]

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!

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^∞?

 

[HallsofIvy]

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.