Analysis I: Lecture 23

In which we prove another form of Taylor’s theorem.

  • Corollary 59 (Integration by parts) Let f, g : [a,b] \to \mathbb{R} be differentiable functions with continuous derivatives.  Then \displaystyle \int_a^b f'g = f(b) g(b) - f(a) g(a) - \int_a^b fg'.  This followed from the product rule and Corollary 57.
  • Theorem 60 (Taylor‘s theorem with an integral form of the remainder) Take n \geq 1, and let f : (-a,a) \to \mathbb{R} be n times differentiable with continuous n^{th} derivative.  For |t| < a, we have \displaystyle f(t) = f(0) + f'(0) t + \frac{f''(0)}{2} t^2 + \dotsb + \frac{f^{(n-1)}(0)}{(n-1)!} t^{n-1} + R_n where \displaystyle R_n = \frac{1}{(n-1)!} \int_0^t (t-x)^{n-1} f^{(n)}(x) \mathrm{d}x.  We proved this by using integration by parts repeatedly.
  • Lemma 61 (Mean value theorem for integrals) Let f:[a,b] \to \mathbb{R} be continuous.  Then there is some  c \in (a,b) such that \int_a^b f = (b-a) f(c).  To prove this, we defined F(t) = \int_a^t f(x) \mathrm{d}x and applied the mean value theorem to it.
  • We used Lemma 61 to obtain Cauchy’s form of the remainder from the integral form.
  • Lemma 62 Let f, g : [a,b] \to \mathbb{R} be continuous, with g(t) > 0 for t \in (a,b).  Then there is some c \in (a,b) such that \int_a^b f(x) g(x) \mathrm{d}x = f(c) \int_a^b g(x) \mathrm{d}xThe idea of the proof is to apply Cauchy’s mean value theorem to the functions H(t) = \int_a^t f(x) g(x) \mathrm{d}x and G(t) = \int_a^t g(x) \mathrm{d}x.
  • We used Lemma 62 to obtain Lagrange’s form of the remainder from the integral form.

Understanding today’s lecture

Can you fill in the details of the proof of Lemma 62?

Can you use the integral form of the remainder in Taylor’s theorem to obtain the Taylor series for the binomial series?  (We did this earlier in the term using Lagrange’s and Cauchy’s forms of the remainder, but it would be good to see how the estimates work out for the integral form.)

Further reading

You might like to have a look at Gareth Taylor’s notes on Taylor’s theorem, with the various forms of the remainder that we’ve seen in this course.  Wikipedia also records several forms of the remainder term.

Preparation for Lecture 24

We have defined the integral for functions defined on a closed, bounded interval.  How might we extend that to define things like \int_a^{\infty} f?  When does this integral converge?

For which \alpha does the integral \int_1^{\infty} x^{-\alpha} \mathrm{d}x converge?  How is this related to \sum\limits_{n=1}^{\infty} n^{-\alpha}?  More generally, how is the convergence of \int_1^{\infty} f(x) \mathrm{d}x related to the convergence of \sum\limits_{n=1}^{\infty} f(n)?

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