Preface
This section gives an introduction to approximations of functions by a rational function of given order.
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Glossary
Padé Approximations
A Padé approximant is the "best" approximation of a function by a rational function of given order  under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by the French mathematician Henri Padé (18631953), but goes back to the German mathematician Georg Frobenius (18491917) who introduced the idea and investigated the features of rational approximations of power series. Henri Eugène Padé, while preparing his doctorate under Charles Hermite, he introduced what is now known as the Padé approximanta rational functions that provides a smallest overall error on the interval [𝑎, b].
Actually, the first rational approximation was obtained in seventh century by the Indian mathematician Bhaskara (c.600 – c.680). He has given the rule for a remarkable approximation of the sine function:
den[x_] = 1 + 445/12122*x^2 + 601/872784*x^4 + 121/16662240*x^6;
si[x_] = num[x]/den[x]
When the power series representation of a function diverges, it indicates the inability of the power series to approximate the function in a certain region. A theorem from complex analysis states that if the Taylor series of a function diverges, then that function has singularities in the complex plane. A Padé approximant is a ratio of polynomials that contains the same information that a truncated power series does. Because the polynomial in the denominator may have roots in the region of interest, the Padé approximant may accurately indicate the presence of singularities.
Given a function f and two integers \( m \ge 0 \quad\mbox{and}\quad n \ge 1, \) the Padé approximant of order [m/n] is the quotient of two polynomials P_{m}(x) and Q_{n}(x) of degrees m and n, respectively:
Example: Consider Padé approximants for cosine functions
Example: Suppose we wish to approximate the solution of the ordinary differential equation
The diagonal sequence of Padé approximants corresponding to the differential equation is
Example: Suppose we wish to approximate the solution of the ordinary differential equation
 For P_{2}^{2}(x), these singularities are at x ≈ ±1.7.

For P
^{3}(x), these singularities are at x ≈ ±1.58.  For P_{4}^{4}(x), these singularities are at x ≈ ±1.5712, and x ≈ ±6.5.
Example: Suppose we wish to approximate the solution of the ordinary differential equation
Finally, we plot the solution along with its {5,5}Padé approximation:
pade[x_] = N[%];
s = NDSolve[{y'[x]==x^2 (y[x])^2, y[0]==1}, y, {x,0,2}]; Plot[{pade[x],Evaluate[y[x]/.s]}, {x,0,2}, PlotRange>All, PlotLabels>{"Pade","true"}] 

Padé approximation along with the rtue solution to the Riccati equation.  Mathematica code 
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 Baker G.A. (1975), Essentials of Padé approximants. Academic, New York
 John P. Boyd, Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, Computers in Physics, Volume 11 Issue 3, May/June 1997 Pages 299303; https://doi.org/10.1063/1.168606
 Gonnet, P., Güttel, S., and Trefethen, L.N., Robust Padé approximation via SVD, SIAM Review, submitted, 2011.
 Stroethoff, K., Bhaskara’s approximation for the Sine, The Mathematics Enthusiast, 2014, Vol. 11, Issue 3, pp. 485492.
 Wazwaz AM (2006) The modified decomposition method and Padé approximants for a boundary layer equation in unbounded domain. Appl Math Comput 177:737–744
 Xu H (2004) An explicit analytic solution for free convection about a vertical flat plate embedded in a porous medium by means of homotopy analysis method. Appl Math Comput 158:433–443
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