Calculate Pade Approximation for f(x)=1-\frac{1}{2}x+\frac{1}{3}x^2-...

[matematikuvol]

Homework Statement

Pade approximation
$$[N/D]=\frac{a_0+a_1x+...+a_Nx^N}{1+b_1x+...+b_Dx^D}$$
With this approximation we approximate Maclaurin series
$$f(x)=\sum^{\infty}_{i=0}c_ix^i=[N/D]+O(x^{N+D+1})$$
How to calculate $$[1/1]$$ for $$f(x)=1-\frac{1}{2}x+\frac{1}{3}x^2-...$$ ?

Homework Equations

$$[N/D]=\frac{a_0+a_1x+...+a_Nx^N}{1+b_1x+...+b_Dx^D}$$
$$\sum^{\infty}_{i=0}c_ix^i=[N/D]+O(x^{N+D+1})$$

The Attempt at a Solution

$$(1+b_1x)(1-\frac{1}{2}x)=a_0+a_1x$$
$$a_0=1$$
$$b_1-\frac{1}{2}=a_1$$

How to calculate $$a_1,b_1$$

In solution
$$[1/1]=\frac{1+\frac{1}{6}x}{1+\frac{2}{3}x}$$

 

[clamtrox]

Write more terms on both sides
$$(1+b_1 x)(1-x/2+x^2/3) = a_0 + a_1 x + \mathcal{O}(x^3)$$
So what happens to the term proportional to $x^2$?