Question

What is the power series for x^x?

Answer 1

Sure, but I want the formula.

Answer 2

\[\LARGE x^x=\sum_{n=0}^\infty a_nx^n\] I want the formula for the coefficients!

Answer 3

\[\LARGE e^x=\sum_{n=0}^\infty \dfrac{x^n}{n!}\]

Answer 4

replace x by xlnx

Answer 5

XD Wow that's so incredibly easy I feel stupid for not noticing that thanks haha!

Answer 6

any other way you have it worked ?

Answer 7

but the power series should not have any x terms in the coefficients an right ?

Answer 8

Nope, nothing interesting going on here like usual, I was really just asking a question this time because I am curious about tetration: http://en.wikipedia.org/wiki/Tetration
Now I guess I should change my question to what I really want, the power series for: \[\LARGE x^{x^x}\]

Answer 9

Yeah, good point, maybe this question still has more to think about. Honestly I am trying to figure out what it means to do tetration a noninteger number of times.

Answer 10

yeah

Answer 11

I suppose we can also find x^x^x by just plugging in \[\LARGE e^{x^x lnx}=\sum \frac{(x^x \ln x)^n}{n!}\] but that's really less satisfying and not quite what I'm looking for.

Also it seems to me that centering the power series at 1 is probably the best idea since 0^0^0 or 0^0 is a little bit questionable but 1^1^1 is a fairly good number. I don't know, just an idea.

Answer 12

idk if we can call it power series when its no longer a polynomial..

Answer 13

Yeah, I agree with you. I'm trying to see if I can find a way to turn these into polynomials. Doesn't look too good, but it's a start.

Answer 14

maybe we need to use double sums...

Answer 15

that's interesting, are there power series that use double sums? I've never heard of that before lol

Answer 16

me neither.. if we represent x^xlnx as another power series, we're done right ?

Answer 17

because polynomials are closed under multiplication.. product of two power series is another power series ?

Answer 18

True that's a good idea! maybe we can sort of multiply them out with some binomial theorem factorials or something and it'll all be fine haha.