Can the Taylor series be inverted without using Lagrange's theorem?

[Nishkin]

How can you invert a Taylor serie?

x=y+Ay^2+By^3+Cy^4...

to y=ax+bx^2+cx^3 ...
without the lagrange theorem... must go from x=y+Ay^2+By^3+Cy^4... to y=ax+bx^2+cx^3 ...


Need help thanks!

 

[quantumdude]

If you know the function from which the Taylor series was derived, then this paper might help:

Nested Derivatives: A simple method for computing series expansions of inverse functions

Basically, if you know f(x) then the method in that paper allows you to find a power series for f^-1(x).

 

[tacman]

Yes, it is possible to invert a Taylor series without using Lagrange's theorem. One way to do this is by using the method of undetermined coefficients. This method involves finding a power series representation for the inverse function and then using the coefficients to determine the values of the inverse function at each point.

To invert the given Taylor series, you can start by writing it in the form:

y = x + Bx^2 + Cx^3 + Dx^4 + ...

Then, by equating the coefficients of each power of x on both sides, you can solve for the coefficients B, C, D, etc. This will give you a power series representation for the inverse function:

x = y + Ay^2 + By^3 + Cy^4 + ...

From this representation, you can then use the coefficients A, B, C, etc. to determine the values of the inverse function at each point.

It is important to note that this method may not always be applicable or may be more complicated for certain Taylor series. In such cases, Lagrange's theorem may be a more efficient way to invert the series. However, it is possible to invert a Taylor series without using Lagrange's theorem, as shown above.