Another maclaurin vs. taylor series question

[eprparadox]

Hey guys,

Struggling with understanding this taylor vs. maclaurin series stuff.

So a few questions. Let's say that we have some function f(x).

1. By saying that we want to find the power series of f(x) and nothing else, are we implicitly stating that we are looking for a maclaurin series or a taylor series? OR do we have to specify around what point we're looking for this power series?

2. I went on wolfram alpha and I found the taylor series of sin(x) at x =2 and the maclaurin series of sin(x) (at x = 0). and then I evaluated the answers at the same x value (x = 4). And I got different answers. I thought they should come out to the same value since we're still expanding the same initial function, sin(x). Ultimately, I'm finding it difficult to understand how these two seemingly different power series are converging to one function.

3. What does it mean for a power series to be centered around a value, other than for it to be the center of the circle of convergence.

Hope these questions made sense. I just want to get a really strong intuition for maclaurin vs. taylor series.

Thanks!

 

[Simon Bridge]

http://en.wikipedia.org/wiki/Taylor_series
A Maclearan series is a Taylor series, but not all Taylor series are Maclearan series.
Both are examples of power series.

Power series: $$f(x)=\sum_{n=0}^\infty a_nx^n$$

Maclearan series: $$f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$

Taylor series:$$f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$$

So:
1. if the type of series is not indicated, then it doesn't matter to the argument being made.
Where there may be confusion, following statements should clear it up.

2. the series expansion for sin(x) has infinitely many terms - you were evaluating two different approximations so you got different answers.

3. see the wikipedia article (above) for details.
The power series is most accurate close to the point x=a in the formula - the series is said to be centered around this value. The Taylor series makes that point explicit while for the general power series it may not be so clear.

 

[eprparadox]

Hey, thanks so much for the response. I'm reading Mary Boas' "Mathematical Guide in the physical sciences" and in it, she states a few theorems on power series. One of them is this:

"The power series of a function is unique, that is, there is just one power series of the form $$f(x)=\sum_{n=0}^\infty a_nx^n$$ which converges to the given function."

So in our case, our function is the sin(x) but there is more than one power series that converges to that function (the maclaurin series and the taylor series). Are they saying there is only one unique power series for any given POINT on the function?

Thanks again for the help

 

[Simon Bridge]

The quote refers to the whole power series. There are many series which will sum to an approximation to the function.

The taylor series about a=4 and about a=0 are two ways of arriving at the same power series.
The steps are different.

 

[CellCoree]

I can provide a response to your questions about maclaurin vs. taylor series.

1. When we say we want to find the power series of a function, it is important to specify the point around which we are expanding the series. This point is known as the center of the series. If the center is at x=0, then it is a maclaurin series. If the center is at a different value, then it is a taylor series. So, to answer your question, we do need to specify the center of the series.

2. It is important to note that the taylor series and the maclaurin series are different representations of the same function. However, they are only equal when the series is centered at the same point. In your example, the taylor series of sin(x) at x=2 and the maclaurin series of sin(x) at x=0 are not equal because they are centered at different points. This is why you are getting different values when evaluating them at the same x value. The taylor and maclaurin series only converge to the same function when they have the same center.

3. A power series centered at a value means that the series is being expanded around that particular value. The center is usually chosen based on the properties of the function and the desired application. For example, if we want to approximate a function near a certain point, we would choose that point as the center of the series.

I hope this helps clarify your understanding of maclaurin vs. taylor series. It is important to remember that they are just different representations of the same function and the choice of center affects the convergence and accuracy of the series.