Science and math experience.

Experiments geared toward discovering that using more and more of the  expansion results in better and better approximation. &nbs p;Halley's way of  calculating accurate decimals of .  Expansions by substitution.  Expansions by differentiation.   ;Expansions by integration.  Recognition of  expansions.  Expansions that satisfy a priori error bounds.

Mathematics.

The expansion of a function in powers of as a file of  polynomials with higher and higher orders of contact with at .  Netwon's method.  Multiplying and di viding expansions.  Using expansions to help calculate limits at a point.  Expansions and the complex exponential function.& nbsp; Using expansions to help to get precise estimates of some integrals.

Science and math experience.

Centering expansions for good approximation.  Newton's method for root finding.  Successes and failures of Newton's method.& nbsp; Using the complex exponential to generate trigonometric identities.  Comparing reflecting properties of spherical mirrors and the reflecting properties of parabolic mirrors.  Using expansions to see why spherical mirrors have limited ability to concentrate light rays.  Behavior of expansions very close to .  Behavior of expansions far away from .

Mathematics.

Taylor's formula for expansions in powers of .

Science and math experience.

Euler, Midpoint and Runge-Kutta approximations of given .   Experiments comparing the quality of Midpoint and Runge-Kutta approximations.  A daption of Euler, Midpoint and Runge-Kutta approximations to approximating the plots of the differential equation , with given.  Taylor's formula in rever se. L'Hospital's rule by dividing the leading term of the expansion of the denominator into the leading term of the expansion of the numerator. Centering the expansion for best approximation.  Experiments comparing the derivative of the expansion and the expansion of   ;the derivative.

Mathematics.

Barriers and complex singularities.  The convergence interval of an expansion as the interval between the barriers.  Why som e functions such as have convergence barriers and others such as and do not.  Why functions such as and do not have expansions in powers of but do have expansions in powers of for .  Why the convergence in tervals for , and are the same.

Science and math experience.

Shortcuts based on the expansion of in powers of .  Using the expansion of in powers of for drug dosing.  Infinite sums of numbers resulting from expansions. Barriers resulting from splines. Infinite sums and decimals.  Experiments relating expansions in powers of to interpolating polynomials.  Rung e's disaster.

Mathematics.

Functions defined by a power series.  Functions defined by power series via differential equations.  The power series conver gence principle, which says that if for some positive number the infinite list
       
is bounded , then the power series       
converges for .