Taylor expansions
What better than a polynomial to approximate, near a point, a complicated function? Better yet: if the latter is "sufficiently regular", the coefficients of its Taylor expansion are its successive derivatives; this is the famous Taylor formula that teaches us this. Yet we get surprises: the adequacy between Taylor expansion at any order and Taylor development is sometimes called into question. Marvels of mathematics, and in particular of analysis, nobody can do without Taylor expansions. They are indeed indispensable in limits of functions, in the study of particular points of a curve, but also in probability calculations.
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Essential tools of analysis
Given a somewhat complicated function, who has not wished they could replace it with a much simpler one that "behaves in the same way," such as a polynomial? Taylor expansions make this possible in almost every case—at least locally!
Some troubling counterexamples
Think you know everything about asymptotic expansions? Here are a few perplexing counterexamples...
Equivalent functions: a tool for calculating limits
Why go to the trouble of finding equivalents for seemingly well-behaved functions? To gain detailed insight into local or asymptotic behavior—and for applications, too! Without these techniques, even spreadsheets would be unable to perform seemingly innocuous calculations.
The origins of function expansions
The first power-series expansions of functions emerged alongside the development of differential and integral calculus in the late 17th century. Truncating them produces Taylor polynomials!
A fine remainder
Replacing the function itself with a Taylor expansion is justified only if this approximation does not alter the calculation of the properties being studied, whether a limit, an upper bound, or something else. The behavior of the remainder is therefore important.
In the service of curves
Through the magic of analytic geometry, the properties of an algebraic formula find visual expression in the corresponding curves. Taylor expansions play an absolutely crucial role in this constant interplay.