- #1

kreil

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## Main Question or Discussion Point

So I'm studying Taylor Series (I work ahead of my calc class so that when we cover topics I already know them and they are easier to study..) and tonight I found a formula for taylor series and maclaurin series, and i used them to prove eulers identity. However, I don't really know much about taylor or maclaurin series. I have a couple questions...

(i) How was the formula [tex]T(x)=\sum_{n=0}^\infty \frac{f^n(a)}{n!}(x-a)^n+R_n[/tex] derived?

(ii) Is there a definite result for the remainder term R

(iii) What exactly is a Taylor series (what is it used for)? I only know that the formula produces a series of functions which, if combined, construct the original function.

(iv) If anyone could offer up some interesting but difficult problems in relation to taylor series, maclaurin series, or power series, I would appreciate it.

Josh

(i) How was the formula [tex]T(x)=\sum_{n=0}^\infty \frac{f^n(a)}{n!}(x-a)^n+R_n[/tex] derived?

(ii) Is there a definite result for the remainder term R

_{n}(I have seen cauchys and lagranges, im just wondering if they are different forms of the same thing or if its still open)?(iii) What exactly is a Taylor series (what is it used for)? I only know that the formula produces a series of functions which, if combined, construct the original function.

(iv) If anyone could offer up some interesting but difficult problems in relation to taylor series, maclaurin series, or power series, I would appreciate it.

Josh

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