Padé approximants of exp(x)

We can organize and present the Padé    approximants in a table like this:

	0	1	2	3
0
1
2
3

The table above shows, in order, Padé’s first approximants. This is a way to present and organize the Padé approximants. We can use the procedure presented in the previous posts to compute the Padé approximants for the exponential function   . Solving systems presented in the previous posts, leads to the following table:

	0	1	2	3
0
1
2
3

Setting x = 1, we obtain the following values:

	0	1	2	3
0
1	—
2
3

The ‘relative error’ is defined as:

Relative errors of Padé approximants    of    evaluated at x = 1, using the exact value    are shown in the table below:

	0	1	2	3
0
1	—
2
3

The Padé approximants exhibit an alternating sign pattern in their relative errors. This indicates that the Padé approximants oscillate around the true value   , approaching it from both sides. In contrast, the Taylor approximations converge monotonically from below.