Taylor Series vs. Padé Approximants: Which is More Effective?

🧮Par d approximants are rational polynomials formed by dividing one polynomial by another, used to approximate functions.

📈Taylor series use repeated differentiation to produce a polynomial approximation of a function about a particular point in x.

🔄Par d approximants can provide a closer approximation to a function for a longer range of x values compared to Taylor series.

⚖️Par d approximants cancel out the tendency of Taylor series to diverge towards positive or negative infinity.

💡To construct an n over m part a approximant, the Taylor series up to order n + m is equated to the approximant, and the coefficients are calculated.

What are par d approximants?

—Par d approximants are rational polynomials formed by dividing one polynomial by another, used to approximate functions.

How do Taylor series work?

—Taylor series use repeated differentiation to produce a polynomial approximation of a function about a particular point in x.

Why are par d approximants sometimes more effective than Taylor series?

—Par d approximants can provide a closer approximation to a function for a longer range of x values compared to Taylor series.

How do par d approximants cancel out the tendency of Taylor series to diverge?

—By dividing one polynomial by another, par d approximants eliminate the tendency of Taylor series to shoot towards positive or negative infinity.

How are coefficients calculated for an n over m part a approximant?

—The Taylor series up to order n + m is equated to the approximant, and the coefficients are solved using a set of simultaneous equations.

00:04Par d approximants are rational polynomials used to approximate functions.

00:34Taylor series use repeated differentiation to produce a polynomial approximation of a function.

01:48Par d approximants provide a closer approximation to a function for a longer range of x values compared to Taylor series.

03:49Par d approximants cancel out the tendency of Taylor series to diverge towards positive or negative infinity.

05:48To construct an n over m part a approximant, the Taylor series up to order n + m is equated to the approximant, and the coefficients are calculated.