Having difficulty figuring out how to determine if a series has a Maclaurin expansion. Anyone think they can clear this up for me?
What do you mean a series has a Maclaurin expansion...a maclaurin expansion is a series centered around 0?
Taylor
$\displaystyle \sum_{n=c}^{\infty}\frac{(x-c)^{n}f^{(n)}(x)}{n!}$
Macluarin
$\displaystyle \sum_{n=c}^{\infty}\frac{(x-0)^nf^{(n)}(x)}{n!}=\sum_{n=0}^{\infty}\frac{x^{n} f^{(n)}(x)}{n!}$ or am I misunderstanding
and if you are asking how many functions have series that are convergent in their Maclaurin series?
The answer is all of them...no matter what the series is it will converge at $\displaystyle x=0$
$\displaystyle \sum_{n=c}^{\infty}\frac{0^{n}f^{(n)}(0)}{n!}=\sum _{n=c}^{\infty}0=0$
convergent
Well my book defines a Maclaurin series as:
$\displaystyle f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n = f(0) + \frac {f\prime (0)}{1!}x +\frac {f\prime\prime (0)}{2!}x^2 + ...$
So for example, one of my questions says:
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. (assume that the f has a power series expansion. Do not show that R_n(x)->0)
then they give 6 problems:
$\displaystyle f(x)=cos ~x$
$\displaystyle f(x)=sin~2x$
$\displaystyle f(x)=e^{5x}$
$\displaystyle f(x)=xe^x$
$\displaystyle f(x)=sinh ~x$
$\displaystyle f(x)=cosh ~x$
If the instructions didn't tell me that I could use a Maclaurin series, then how would I know?
Let me put it this way...all functions have power series...how many are well known and unmessy? about thirty or so...if it doesnt specify I would use Maclaurin considering the scarcity of taylor series after the initial advent of it....do you need help on any of these series?
Well, not in particular, I'll explain why I'm asking.
In class, we covered Maclaurin series, then suddenly jumped to vectors (we never got to Taylor series, I think we covered maybe a third of this section). In class, I asked my instructor if every function had a Maclaurin series. (to be honest, I don't really understand how it works, I looked at the graphics for Taylor series such as on Wikipedia, and the graphics make a lot of sense, but I don't really understand how the math relates to the graphic.) My instructor got back to me after the next class, and explained when they do have one, but he had to get to a meeting, and so the explanation was very abbreviated, and I didn't really get it. But my instructor has in the past put problems on the test specifically that were asked to him, such as the integral of sec^3(x). So I am concerned that on the test he will put a question such as "Find the Maclaurin series of f(x)" but the function will not have one. If I am not able to distinguish between functions which do and do not have one, I will lose a lot of points and waste a lot of time. So I figured I'd ask how to determine whether a function has one or not.