Favorite Theorem #2: Taylor’s Theorem

Taylor’s Theorem

I am likely lose too many of my regulars with this one but I have to throw it in since it ranks, by far, at the top of my list of fave’s.

Thm: Let f \in C^{k+1}(I) for some interval I \subset \mathbb{R}. For a \in I, we have that

f(a+h) = \displaystyle \sum_{j=0}^k \frac{f^{(j)}(a)}{j!} h^j \ + R_{a,k}(h)
where
R_{a,k}(h) = \displaystyle \frac{h^{k+1}}{k!} \int_0^1 (1-t)^k f^{(k+1)} (a+th) \; dt.

Taylor’s theorem gives a polynomial approximation to a smooth function near a single point. Since approximation is one my favorite topics, Taylor’s theorem is close to my heart. See tomorrow’s spiritual principle for why approximation is so near and dear.

This version of the theorem represents one form of the remainder for the Taylor’s Polynomial approximation to a smooth function. I first came across it in my Calculus III class as an undergrad. I fell in love with it during my first Numerical Analysis class in grad school. I used time and time again in the development of my understanding of discrete approximation methods for differential equations. It was there that I found my topic for a doctoral dissertation . . . . (lost in sentimental thought)

I just reread the paragraph and I must say, it sounds like Taylor’s Theorem is the closest I’ll ever come to having a mistress!!?!! I could do worse.

4 thoughts on “Favorite Theorem #2: Taylor’s Theorem

  1. That’s approximating with polynomials. It’s a much more sordid crowd of functions that I actually approximating.

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