Analysis I: Lecture 13

In which we try to approximate functions by polynomials.

Theorem 28 (Taylor‘s theorem with Lagrange‘s form of the remainder) Let $f : [a,a+h] \to \mathbb{R}$ be a function such that $f$ and its first $n-1$ derivatives are continuous on $[a,a+h]$ and $f$ is $n$ times differentiable on $(a,a+h)$ . Then there is some $\theta \in (0,1)$ such that $\displaystyle f(a+h) = f(a) + h f'(a) + \frac{h^2}{2} f''(a) + \dotsb + \frac{h^{n-1}}{(n-1)!}f^{(n-1)}(a) + \frac{h^n}{n!} f^{(n)}(a+\theta h)$ . We proved this by applying Rolle‘s theorem $n$ times to a suitable auxiliary function.
Theorem 29 (Taylor’s theorem with Cauchy‘s form of the remainder) Let $f : [0,h] \to \mathbb{R}$ be a function such that $f$ and its first $n-1$ derivatives are continuous on $[0,h]$ and $f$ is $n$ times differentiable on $(0,h)$ . Then there is some $\theta \in (0,1)$ such that $\displaystyle f(h) = f(0) + h f'(0) + \frac{h^2}{2} f''(0) + \dotsb + \frac{h^{n-1}}{(n-1)!} f^{(n-1)}(0) + \frac{h^n}{(n-1)!} (1-\theta)^{n-1} f^{(n)}(\theta h)$ . We proved this by applying the Mean value theorem to a suitable auxiliary function.

Understanding today’s lecture

A good way to understand the proofs would be to work through them for particular small values of $n$ . You could try experimenting with other functions that you know about (we’ll meet standard functions such as exponentials and trig functions formally in lectures during the next couple of weeks).

Preparation for Lecture 14

Can you finish our analysis of the binomial example ( $f(x) = (1+x)^r$ where $r$ is rational)?

Remind yourself of the definition of differentiability for functions from $\mathbb{C}$ to $\mathbb{C}$ . At which points is the complex conjugate function $f(z) = \bar{z}$ differentiable?

If I give you a convergent series $\sum\limits_{n=1}^{\infty} a_n z^n$ (where $z \in \mathbb{C}$ ), what can you say about $\sum\limits_{n=1}^{\infty} a_n w^n$ for $|w| < |z|$ ? If I give you a series $\sum\limits_{n=1}^{\infty} a_n z^n$ that does not converge, what can you say about $\sum\limits_{n=1}^{\infty} a_n w^n$ for $|w| > |z|$ ?

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5 Responses to “Analysis I: Lecture 13”

theoremoftheweek Says:
February 15, 2013 at 12:30 pm
As a student pointed out to me at the end of the lecture, in the proof of Theorem 28 I didn’t explicitly say why we were allowed to differentiate the auxiliary function g n-1 times at 0. If you check the definition of g, you’ll see that we’re allowed to do this because f is (n-1) times differentiable at a. Hope that helps to clarify that point.
guest Says:
February 15, 2013 at 2:29 pm
Hi,
just to make sure that I understand the conditions of Taylor’s Thm:
n-times differentiability would _not_ be enough since it only implies continuity of the derivatives on (a,a+h). But in order to prove it by Rolle’s Thm we need to impose the slightly stronger condition that all derivatives are also continuous on the endpoints, correct?
apgoucher Says:
February 15, 2013 at 3:45 pm
This gets more exciting on the complex plane, where the existence of a complex derivative implies the existence of infinitely many complex derivatives (c.f. ‘holomorphic function’) and thus a Taylor series at every point.

Challenge: Find a function $f:\mathbb{R} \rightarrow \mathbb{R}$ , which is infinitely differentiable at every point, and every derivative $f^{(r)}(0) = 0$ , yet the function is non-constant. Note that this construction can’t generalise to the complex numbers.

Due to the propinquity of St. Valentine’s Day, I’ll take this opportunity to say that I $\epsilon >$ analysis!
theoremoftheweek Says:
February 15, 2013 at 6:07 pm
guest: yes, that’s right, the n times differentiability was only on the open interval, and we need continuity of the first n-1 derivatives at the endpoints too. Good questions to ask yourself!

apgoucher: we’ll talk a bit more about complex differentiable functions soon!
Analysis I: Lecture 23 | Theorem of the week Says:
March 11, 2013 at 12:34 pm
[...] used Lemma 61 to obtain Cauchy’s form of the remainder from the integral [...]

Theorem of the week