Questions regarding Taylor series

[11thHeaven]

We just had a lecture on power series today (Taylor and McLaurin's) and I had a couple of questions:

What does it mean for an expansion to be "around the origin"? I thought that the expansion provided an approximation to the original function at all points for which the function was defined.

Similarly, what does it mean for an expansion to be "around a point"? Is this point on the x-axis? Seeing as some functions are defined around the whole number line, how can they be expanded around, say, 5 or 6?

I apologise if this is in the wrong section; I didn't post it in Homework because it's concepts rather than specific problems that I need help with.

Thanks for any help :)

 

[mathman]

Maclaurin series: ∑a_nxⁿ
Taylor series: ∑a_n(x-c)ⁿ

c is the point ("around the point"). Note that Maclaurin series, c = 0, i.e. series around the origin.

 

[tiny-tim]

two sums ∑ a_nxⁿ and ∑ b_n(x-C)ⁿ can be equal, but of course the a_ns and b_ns will be different

{a_n} are the coefficients for an expansion about 0, {b_n} are the coefficients for an expansion about C

 

[11thHeaven]

Yes, but what does it mean for an expansion to be "about a point"? Surely the expansion is for the entire range of the original function?

 

[Mark44]

Yes, but what does it mean for an expansion to be "about a point"? Surely the expansion is for the entire range of the original function?

"About a point" has to do with what the terms in the series look like.

A Taylor series is a series of powers of x - a.
A Maclaurin series is a series in powers or x - 0 (in other words, powers of x). From this, it should be obvious that a Maclaurin series is a special kind of Taylor series.

A particular Taylor series won't converge to the function that produced the series if you get beyond the interval of convergence. As you get a little farther in your studies of series, you'll be working with how to determine convergence intervals.