Taylor Series Tips: Learn & Understand Power Series

[toni]

I really need some tips on taylor series...Im trying to learn it myself but i couldn't understand what's on the book...

Can anyone who has learned this give me some tips...like what's the difference between it and power series (i know it's one kind of power series), why people develop it, and is there any standard way to prove that a function can be represent by a particular taylor series?

Thank you soooo much!

 

[HallsofIvy]

A Taylor series is just a power series calculated in a particular way. Not only is it true that a Taylor series is a type of power series, but if a power series is equal to a function, it must be the Taylor series for that function.

That means I can calculate the Taylor series for, say, f(x)= 1/(1-x), at x= 0, in two different ways:
Using the definition, find the derivatives, evaluate at x= 0, and put those into the formlula: f(0)= 1, f'(x)= (1- x)⁼² so f'(0)= 1, f"= 2(1-x)^-3 so f"(0)= 2, ..., f⁽ⁿ⁾(x)= n!(1-x)ⁿ so f<sup>n[/sub](0)= n! and therefore,
$$\sum \frac{f^{(n)}(0)}{n!}x^n= \sum x^n$$

Or just recall that the sum of a geometric series, $\sum ar^n$ is 1/(1- r). Since 1/(1-x) this must be a geometric series with a= 1 and r= x: That gives
$$\frac{1}{1-x}= \sum x^n$$
just as before. Because they are power series converging to the same function, they musst be exactly the same.</sup>

 

[mathman]

The description given by Halls of Ivy is a special case of Taylor, call MacLauren (sp?) series. In general Taylor series involve powers of (x-a) where a is an arbitrary constant.

 

[toni]

is "a" arbitrary? "a" is also said to be the "center" right?

 

[HallsofIvy]

If you mean the center of the interval of convergence, yes.