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?