Mathematical Recreational Notes – Taylor Series

Standard

I have been revisiting basic mathematical datums for a while and I thought that taylor series was pretty awesome revelation on the part of anyone who thought that a function’s properties can be readily realized once its infinite series form could be observed from its derivatives. Maclaurin series is the name given when the same taylor series is written at x=0. The recurrence of a word “analytic” in this space enthralled me to question that why is this notion so important in case of convergent series. I found some revealing facts for myself to retrospect on how did I study them previously – WEIRD !!

Analytic functions are those that can be given by a convergent power series at the concerned points. Now that the function should be infinitely differentiable and it has to be equal to its taylor series in neighborhood of every point which is a necessary condition.

A function ƒ defined on some subset of the real line is said to be real analytic at a point x if there is a neighborhood D of x on which ƒ is real analytic. Consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer. The function f is said to be of class Ck if the derivatives f’f”, …, f(k) exist and are continuous (the continuity is automatic for all the derivatives except for f(k)). The function f is said to be of class C, or smooth, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth and if it equals its Taylor seriesexpansion around any point in its domain.

If ƒ is an infinitely differentiable function defined on an open set D \subset \mathbb{R}, then the following conditions are equivalent.

1) ƒ is real analytic.
2) There is a complex analytic extension of ƒ to an open set G \subset \mathbb{C} which contains D.
3) For every compact set  K \subset D  there exists a constant C such that for every  x \in K and every non-negative integer k the following estimate holds:
 \left | \frac{d^k f}{dx^k}(x) \right | \leq C^{k+1} k!
Now Taylor series can be written for complex functions too but that’s the part of complex analysis.  When the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.Now let me give you some famous taylor series (maclaurin series at x=0) –
\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\quad\text{ for all } x\!
\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\quad\text{ for all } x\!
In the above series, there is a recognizable pattern but is it so obvious for tan(x) and sec(x)?
\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots\quad\text{ for }|x| < \frac{\pi}{2}\!
where the Bs are Bernoulli numbers.
\sec x = \sum^{\infin}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}\quad\text{ for }|x| < \frac{\pi}{2}\!
The Ek in the expansion of sec(x) are Euler numbers.
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