At first I hoped the sum gives something common and simple, but now I have the same opinion. Even if it gives the theta function, I wouldn't be able to calculate further with it.
Numerical evaluation doesn't give you understanding, you just can compare whether two calculations give the same...
Thank you for the hint. I was thinking about it, and I looked at other Jacobi theta functions (e.g. here: http://mathworld.wolfram.com/JacobiThetaFunctions.html), but I still don't know how to use it. The z parameter of the \vartheta_3 function can be a complex number, which is good as I can use...
I don't know how to. The power operation is not anyhow associative, so I'm unable to transfer x^(2^n) to anything like (y^2)^n. I've already tried to apply exp and log, but without success.
That's one of the Taylor series lists I found. When decomposing combinations of several such functions, I still get only a polynomial grows of exponents, not exponential.
Well, I'm trying to compare the recessional velocity formula for the Doppler effect redshift and for the cosmological redshift. While the formula
v_{rec} = c \frac{λ_{obs} - λ_{rest}}{λ_{rest}}
is easy to derive for the Doppler redshift, the maths for the cosmological redshift is much more...
You have to convert the upper fractions to a common denominator. Then you'll find that the numerator cancels out with part of the denominator. You'll get (I hope :smile:)
\frac{b^2+b+ab+a}{b^2-ab-2a^2}.
I need to calculate \sum_{n=0}^{∞}x^{(2^n)} for 0≤x<1. It doesn't resemble any basic taylor series, so I have no idea how to sum it up. Any hint, or the resulting formula?
This series comes from a physical problem, so I suppose (if I didn't make a mistake) that the series is sumable, and...