What are some sums of infinite series that are = to 'e'?

[mesa]

We all know about the sum of the infinite series,

1 + 1/1! + 1/2! + 1/3! + ... to 1/inf! = e

What other series do we have that are equal to 'e'?

 

[Panphobia]

http://en.wikipedia.org/wiki/List_of_representations_of_e

 

[mesa]

http://en.wikipedia.org/wiki/List_of_representations_of_e

I honestly don't know how you guys find so much more on Wikipedia than I do...

Either way, that is an excellent list of 'infinite series', are there any others?

 

[Mandelbroth]

I honestly don't know how you guys find so much more on Wikipedia than I do...

Either way, that is an excellent list of 'infinite series', are there any others?

I like $$e=\sum_{n\geq 0}f(n),$$ where $\displaystyle f(n)=\left\{\begin{matrix}e & \text{if } n=0 \\ 0 & \text{if } n\neq 0\end{matrix}\right.$.

 

[mesa]

I like $$e=\sum_{n\geq 0}f(n),$$ where $\displaystyle f(n)=\left\{\begin{matrix}e & \text{if } n=0 \\ 0 & \text{if } n\neq 0\end{matrix}\right.$.

That's different... :)

 

[HallsofIvy]

I like $$e=\sum_{n\geq 0}f(n),$$ where $\displaystyle f(n)=\left\{\begin{matrix}e & \text{if } n=0 \\ 0 & \text{if } n\neq 0\end{matrix}\right.$.

Personally, I prefer $$e=\sum_{n\geq 0}f(n),$$ where $\displaystyle f(n)=\left\{\begin{matrix}e & \text{if } n=1 \\ 0 & \text{if } n\neq 1\end{matrix}\right.$

 

[mesa]

So is this all of them, as in if someone found something else outside of these then it's new to maths?

 

[Mandelbroth]

So is this all of them, as in if someone found something else outside of these then it's new to maths?

Do you honestly think we know all of mathematics?

I'd venture to guess that there are at least SOME (*cough*SUM*cough*) other representations out there.

 

[mesa]

Do you honestly think we know all of mathematics?...

Nope, just trying to see if someone else has more information. Do you have recommendations on where to find out more on the subject such as books, websites, etc.?

 

[Jorriss]

Nope, just trying to see if someone else has more information. Do you have recommendations on where to find out more on the subject such as books, websites, etc.?

That list is not exhaustive. There is, as far as I know, not much interest in simply finding new infinite series that add up to 'e,' except perhaps if a new series converges much more rapidly than a previously known series.

 

[mesa]

That list is not exhaustive. There is, as far as I know, not much interest in simply finding new infinite series that add up to 'e,' except perhaps if a new series converges much more rapidly than a previously known series.

I would imagine that would be valuable from a computing standpoint. What's the current record, 10 trillion decimal places or something to that effect?

 

[Jorriss]

I would imagine that would be valuable from a computing standpoint. What's the current record, 10 trillion decimal places or something to that effect?

In what sense valuable? I do not believe there is much interest in finding more decimal places to e beyond the novelty of having found new decimals - though someone more knowledgeable can certainly correct me.

 

[Stephen Tashi]

Do you have recommendations on where to find out more on the subject such as books, websites, etc.?

Why not concentrate on finding infinite series that sum to 1 and then multiply them by e? That would let you find series that sum to \pi or 42 or whatever.

 

[mesa]

In what sense valuable? I do not believe there is much interest in finding more decimal places to e beyond the novelty of having found new decimals - though someone more knowledgeable can certainly correct me.

I find it remarkable we live in a time where we know a 1,000,000,000,000+ decimal places of 'e'. Euler and the like would probably be in awe of such an accomplishment.

Also I think figuring out new ways of calculating 'e' is important since it could help us gain new insight and lead to better maths in the future (much like it has proven to do so in the past).

 

[mesa]

Why not concentrate on finding infinite series that sum to 1 and then multiply them by e? That would let you find series that sum to \pi or 42 or whatever.

Brilliant!

$$e=e*2\sum_{n= 2}^{∞}(n-1)/((2)n!)$$
$$42=42*2\sum_{n= 2}^{∞}(n-1)/((2)n!)$$
$$pi=pi*2\sum_{n= 2}^{∞}(n-1)/((2)n!)$$
$$llama=llama*2\sum_{n= 2}^{∞}(n-1)/((2)n!)$$

Yes I 'cheated' with the 2 multiplier but so far I have only been able to derive an infinite sum to 1/2 and finals are calling...
These things fun!