Series

glebovg · **Joined:** Mon, 28 Dec 2009 00:16:28 UTC **Posts:** 228

How to find series representation for quantities such as $\pi e$ , $\frac{\pi }{e}$ , ${\pi ^\pi }$ , and ${e^e}$ ?

Shadow · **Posted:** Mon, 9 Jul 2012 04:53:49 UTC

glebovg wrote:

How to find series representation for quantities such as $\pi e$ , $\frac{\pi }{e}$ , ${\pi ^\pi }$ , and ${e^e}$ ?

$e^e=\sum{e^n\over n!}$

$\pi^\pi=\sum {(\pi\log\pi)^n\over n!}$

For $\pi e$ use the Cauchy rule for multiplication of power series and pick your favorites for $\pi$ and

glebovg · **Joined:** Mon, 28 Dec 2009 00:16:28 UTC **Posts:** 228

Shadow wrote:

glebovg wrote:

How to find series representation for quantities such as $\pi e$ , $\frac{\pi }{e}$ , ${\pi ^\pi }$ , and ${e^e}$ ?

$e^e=\sum{e^n\over n!}$

$\pi^\pi=\sum {(\pi\log\pi)^n\over n!}$

For $\pi e$ use the Cauchy rule for multiplication of power series and pick your favorites for $\pi$ and

How did you come up with ${\pi ^\pi }$ ?

Shadow · **Posted:** Mon, 9 Jul 2012 05:16:02 UTC

glebovg wrote:

Shadow wrote:

glebovg wrote:

How to find series representation for quantities such as $\pi e$ , $\frac{\pi }{e}$ , ${\pi ^\pi }$ , and ${e^e}$ ?

$e^e=\sum{e^n\over n!}$

$\pi^\pi=\sum {(\pi\log\pi)^n\over n!}$

For $\pi e$ use the Cauchy rule for multiplication of power series and pick your favorites for $\pi$ and

How did you come up with ${\pi ^\pi }$ ?

Just modified the basic e^x

by recalling $\pi=e^{\log\pi}$ , a simple parlor trick.

glebovg · **Joined:** Mon, 28 Dec 2009 00:16:28 UTC **Posts:** 228

Also, I do not think the Cauchy product would give me the explicit form for $\pi e$ and $\frac{\pi }{e}$ .

Shadow · **Posted:** Mon, 9 Jul 2012 05:33:36 UTC

glebovg wrote:

Also, I do not think the Cauchy product would give me the explicit form for $\pi e$ and $\frac{\pi }{e}$ .

Very well, here are some other versions:

$\pi e=\sum{(1+\log\pi)^n\over n!}$

${\pi\over e}=\sum{(\log\pi -1)^n\over n!}$

glebovg · **Joined:** Mon, 28 Dec 2009 00:16:28 UTC **Posts:** 228

How do you generally come up with series representations, say for $\pi$ ?

I know Ramanujan came up with a lot of formulas for $\pi$ and other quantities without any derivations, but how would you derive series for quantities such as $\pi$ ?

Shadow · **Posted:** Mon, 9 Jul 2012 05:50:45 UTC

glebovg wrote:

How do you generally come up with series representations, say for $\pi$ ?

I know Ramanujan came up with a lot of formulas for $\pi$ and other quantities without any derivations, but how would you derive series for quantities such as $\pi$ ?

Look for functions that have pi as an argument or output that is easily found and get their taylor series.

outermeasure · **Posted:** Mon, 9 Jul 2012 06:01:02 UTC

glebovg wrote:

Also, I do not think the Cauchy product would give me the explicit form for $\pi e$ and $\frac{\pi }{e}$ .

Why not?

For example,
$\begin{aligned} \pi&=\sum_{n\in\mathbb{N}} \frac{(n!)^2 2^{n+1}}{(2n+1)!}\\ e&=\sum_{n\in\mathbb{N}} \frac{1}{n!}\\ e^{-1}&=\sum_{n\in\mathbb{N}} \frac{(-1)^n}{n!} \end{aligned}$
are absolutely convergent series, so taking products,
$\begin{aligned} \pi e&=\sum_{(m,n)\in\mathbb{N}^2}\frac{(n!)^2 2^{n+1}}{(2n+1)!m!}\\ \frac{\pi}{e}&=\sum_{(m,n)\in\mathbb{N}^2}\frac{(-1)^m(n!)^2 2^{n+1}}{(2n+1)!m!} \end{aligned}$
are series representation of $\pi e$ and $\pi/e$ .

outermeasure · **Posted:** Mon, 9 Jul 2012 06:02:02 UTC

Shadow wrote:

glebovg wrote:

How do you generally come up with series representations, say for $\pi$ ?

I know Ramanujan came up with a lot of formulas for $\pi$ and other quantities without any derivations, but how would you derive series for quantities such as $\pi$ ?

Look for functions that have pi as an argument or output that is easily found and get their taylor series.

I think Ramanujan was looking at hypergeometric functions or suchlikes.

glebovg · **Joined:** Mon, 28 Dec 2009 00:16:28 UTC **Posts:** 228

Thanks for help.

Shadow · **Posted:** Mon, 9 Jul 2012 06:09:10 UTC

outermeasure wrote:

Shadow wrote:

glebovg wrote:

How do you generally come up with series representations, say for $\pi$ ?

I know Ramanujan came up with a lot of formulas for $\pi$ and other quantities without any derivations, but how would you derive series for quantities such as $\pi$ ?

Look for functions that have pi as an argument or output that is easily found and get their taylor series.

I think Ramanujan was looking at hypergeometric functions or suchlikes.

Yes, he beat that horse to death a lot.

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