# Packages for interval and radius of power series

What packages exist for studying power series(not sure the plural)?

I’m trying to find the formula that originated the series, as well as the radius and interval of convergence.

For finding the Taylor series for a given function at a given point, check out TaylorSeries.jl

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In what form do you have the series? Any finite list of leading coefficients defines a polynomial with infinite radius of convergence, so you really need some way to construct arbitrary coefficients, or at least a way to show they don’t become zero eventually, before you can investigate radius. If the coefficients are integers, the best tool for series identification is often the (non-Julian) Online Encyclopedia of Integer Sequences <oeis.org>

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Thank’s does it also find the radius of convergence?

``````function f(n)
s = 0.0
for k = 1:n
s += 1/(k^2-1)*(x-2)^k
end
return s
end
``````

I want to be able to solve for when the sum converges.

If you have a formula for the coefficients as a function of n then you can use the Cauchy ratio criterion etc.

I think there’s a typo in your formula. The `k=1` term has a division by `(1^2-1)` and there’s some ambiguity about the role of `x` here.

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And ** is not Julia syntax. And presumably the power of x is k not n?

that’s what I thought. I’m tring to represent a series like:

``````f(x)=sum((x^n)/n) for the limit of n-> infinity,
``````

so that I can run a test like

``````f(x+1)/f(x)<1
``````

It sounds like you need symbolic manipulation for that, with an explicit formula for the nth coefficient.

that was meant to be a power series, not a taylor series, but yeah. I’ve been using ModelingToolkit, but integrals aren’t developed yet. Since it’s shown as a sum, if there is a way to test a sum at n=infinity, then I should be able to find x s.t. f(x)<1, using Cauchy.

x is a variable that I want to manipulate, such that f(x)<1.

I changed it to k. the ‘**’ was a typo meant to be ‘*’ not ‘^’.

I don’t know a way to do the asymptotic limit k -> Inf using ModelingToolkit; you’ll have to do that part by hand.

Ok, thanks. I can do that. I didn’t think ModelingToolkit had that. I was just trying to see if there was a way that I could get at the stuff inside a sum, or use a limit tool like forward diff.

I’m also not sure what to use to test inequalities

f(x)<1 for example

Again, if you want an analytical result you will need to e.g. solve f(x) = 1 analytically.
Otherwise you can, in principle, use e.g. IntervalConstraintProgramming.jl to solve the inequality numerically.

this looks promissing. The use of interval arithmetic would be good. Someone else suggested using Interval Arithmetic with Roots to find the domain of functions.

``````using Plots
c(n,x)=(x^n)/((2*n)-1)
u(x)=c(Inf,x)*(x-0)^Inf
plot(u)
``````

I’m able to plot the area where r converges (u(x))<1, I’m not sure how to show the domain of this. I think it can be done with IntervalArithmetic.jl.

Could Richardson.jl be used to find the limit of k-> infinity)?

I have never used it but it looks like that’s what it’s designed for.

[Please make sure to copy and paste the names of packages correctly always, and preferably provide links.]

done!