Confused about Taylor and Maclaurin Series

[paul2211]

Currently, I'm doing some self studying on series, and I'm a bit confused regarding c (the value that the series is expanded about).

For example, does the Maclaurin series expansion of Sin(x) and the Taylor series of Sin(x) about c = 1 both converge to Sin(x)?

If so, what does the value of c do in this case? Can someone explain to me of its significance?

If they are not equal, does the Taylor series add up to Sin(x-1)? If this is the case, then what is the difference between a Taylor series of Sin(x) about c = 1 vs. a Maclaurin series of Sin(x-1)?

I'm really confused about this matter right now, and I hope I made my question clear.

Thank you guys in advance, and I'll be really grateful if someone can clear this up for me.

 

[austintrose]

There's not really a difference between a Taylor Series and a MacLaurin Series. Rather, the MacLaurin series is just a special case of the Taylor series. It's a MacLaurin series if it's centered around 0, (c = 0)

The "c" that you see in the Taylor series is simply a horizontal shift of the functions. Exactly like you you would translate any old function.

I.e., the Taylor Series for sin(x), centered at five, would approach the function sin(x-5).

 

[lurflurf]

Yes for sin(x) the expansion about any point converges to sin(x). For other functions this might not be the case. The value c is the center of expansion. The idea is we are using information about the function near c to tell us what the function does further away. Think about a polynomial
3+4(x-3)-5(x-3)^2+6(x-3)^3-7(x-3)^4+8(x-3)^5
and
-2727+4192 x-2597 x^2+810 x^3-127 x^4+8 x^5
are the same polynomial but the first form is centered about x=3 and the second is centered about x=0. It is like two ways of looking at the same information.

 

[alexfloo]

If a series converges, it will converge on an interval that has 'c' at its center. In the case of sin, this interval covers the whole real line, regardless of c, so the two series are equal.z

 

[mathman]

The main difference between the two is simply their definitions. Maclaurin series are power series around 0, while Taylor series are expansions around any point. In other words, Maclaurin series are special cases of Taylor series.