Get the closed form of Taylor series with Maple

Question

Is it possible to get the closed form of Taylor series with Maple?

The series command can give any given number of terms, but the question is about the closed form of the entire Taylor series.

Answer 1

Try convert(expression, FPS, variable) or convert(expression, FPS, variable = a), e.g.

convert(exp(x), FPS, x);

$$ \sum_{k=0}^\infty \dfrac{x^k}{k!} $$

Of course it won't always work, but for many functions that have reasonably simple Taylor (or Laurent) series it will.

EDIT: This might sometimes work for a bivariate series as well. Maybe not in your case, but in the simpler case of $(1 - a x - b y)^{-1}$:

convert((1-a*x-b*y)^(-1), FPS, x);
convert(%, FPS, y);

$$ \sum _{k=0}^{\infty } \left( \sum _{{\it k1}=0}^{\infty }{\frac {{a}^{ k}{x}^{k}{b}^{{\it k1}}{\it pochhammer} \left( k+1,{\it k1} \right) {y }^{{\it k1}}}{{\it k1}!}} \right) $$