Taylor Series Tips: Learn & Understand Power Series

In summary, A Taylor series is a type of power series that is calculated in a particular way. It can be used to represent a function if the power series is equal to that function. The Taylor series for a function can be calculated in two ways: using the definition by finding derivatives and evaluating at a particular point, or by recognizing it as a geometric series. The center of a Taylor series is an arbitrary constant.
  • #1
toni
19
0
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!
 
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  • #2
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 fn[/sub](0)= n! and therefore,
[tex]\sum \frac{f^{(n)}(0)}{n!}x^n= \sum x^n[/tex]

Or just recall that the sum of a geometric series, [itex]\sum ar^n[/itex] is 1/(1- r). Since 1/(1-x) this must be a geometric series with a= 1 and r= x: That gives
[tex]\frac{1}{1-x}= \sum x^n[/tex]
just as before. Because they are power series converging to the same function, they musst be exactly the same.
 
  • #3
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.
 
  • #4
is "a" arbitrary? "a" is also said to be the "center" right?
 
  • #5
If you mean the center of the interval of convergence, yes.
 

1. What is a Taylor series?

A Taylor series is a representation of a function as an infinite sum of terms, where each term is calculated based on the derivatives of the function at a specific point. It is used to approximate a function and can be used to find the value of the function at any point within its interval of convergence.

2. Why are Taylor series important?

Taylor series are important because they allow us to approximate complex functions with simpler ones, making it easier to analyze and understand them. They are also used in various fields of science, such as physics and engineering, to model physical phenomena.

3. What is the process of finding a Taylor series?

The process of finding a Taylor series involves finding the derivatives of the function at a specific point, plugging them into the formula for the Taylor series, and simplifying the resulting terms. The final series will depend on the point chosen and the number of terms included in the calculation.

4. What is the difference between a Maclaurin series and a Taylor series?

A Maclaurin series is a type of Taylor series where the point chosen for the calculation is 0. In other words, it is a special case of a Taylor series. Maclaurin series are often used for functions that are symmetric about the origin, while Taylor series can be used for any point within the interval of convergence.

5. How can I use Taylor series to approximate a function?

To use Taylor series to approximate a function, you need to choose a point within the interval of convergence and include a sufficient number of terms in the series. The more terms you include, the more accurate the approximation will be. You can then plug in a value for the variable in the series to find an approximation for the function at that point.

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