# Rudin 5.18 (Trick guidance)

1. Mar 4, 2009

### Quantumpencil

1. The problem statement, all variables and given/known data

Suppose f is a real function on [a, b], n is a positive integer, and $$\f^{(n-1)}$$
exists for every t in [a, b]. Let $$\alpha,\beta$$, and P be as in Taylor’s theorem
(5.15). Define

$$\ Q(t) = \frac{f(t)-f(\beta)}{t-\beta}$$

for $$\ t \in [a, b], t \neq \beta$$,

differentiate

$$\ f(t)-f(\beta)=(t-\beta)Q(t)$$

n − 1 times at $$\ t = \alpha$$, and derive an alternate Taylor’s theorem:

$$\ f(\beta)=P(\beta)+\frac{Q^{(n-1)}(\alpha)}{(n-1)!}(\beta-\alpha)^{n}$$ (I had to put this here to make the above expression stay on one line)

2. Relevant equations

3. The attempt at a solution

So first I did the differentiation n-1 times, you notice a pattern and since f(beta) is constant, you get

$$\ f^{(n}}(t)= nQ^{(n-1)}+(t-\beta)^nQ^{(n)}(t)$$

Then from Taylor's theorem we know that

$$\ f(\beta) = P(\beta) + \frac{f^{(n)}(x)}{n!}(\beta - \alpha)^{n}$$

Just plugging in that expression into Taylor's theorem is real damn close to the result I need. How do I get rid of the extra Q^{(n)} in the numerator? (or is my differentiation wrong and I'm not catching it?)

Thanks a mil guys.

Last edited: Mar 4, 2009
2. Mar 4, 2009

### Avodyne

Both your equations in part (3) are wrong.

3. Mar 4, 2009

### Quantumpencil

The second equation is Rudin 24... I don't see how it could be wrong -_-.

The first is just differentiating the given expression. We'd use the product rule (n-1) (I think there I used it n times), and that would be the result yes?