1. Taylor Series (4/7/06) We have seen that if f(x) is a function for which we can compute all of its derivatives (i.e., first derivative f '(x), second derivative f (2) (x), third derivative f (3) (x), and so on), then we can write down a representation of f as a power series in terms of the values of those derivatives at the center of the interval of convergence. This is called the Taylor Series for f(x).

2. Taylor Series centered at 0 (aka Maclaurin Series) We get that, on the interval of convergence, f(x) = f(0) + f '(0) x + f (2) (0) x 2 / 2! +… + f (n) (0) x n / n! +… = To estimate f, we can use the first so many terms (as a calculator does!).

3. Examples of series centered at 0 Since if f (x) = e x, then for every n, f (n) (x) = e x also, so f (n) (0) = 1. Hence the Taylor Series for e x is very simple: e x = 1 + x + x 2 /2! + … + x n /n! + …, which converges everywhere. Show that the Taylor series for sin(x) is sin(x) = x – x 3 /3! + x 5 /5! – x 7 /7! + …. By computing the derivatives of sin at 0.

4. Taylor Series centered at a If it is easier (or more useful) to evaluate f ‘s derivatives at a point a rather than zero, then we simply center the power series at a and evaluate the derivative at a: f(x) = f(a) + f '(a) (x-a) + f (2) (a) (x-a) 2 / 2! + …+ f (n) (a) (x-a) n / n! +…

5. Examples of series centered at 1 The natural log function isn’t even defined at 0, so we can’t center a series for it there. Show, computing derivatives evaluated at 1, that a Taylor series for ln(x) is ln(x) = (x-1) - (x-1) 2 /2 + (x-1) 3 /3 - (x-1) 4 /4 + … (Note that this only converges on 0 < x  2) Now try it for the square root function. These derivatives at 1 are a bit trickier!

6. Assignment For Monday, read Section 11.10 through page 767 and do Exercises 3, 5, 7, 11, 14, & 15. Wednesday will be an optional Q & A session. Friday (4/14) is Test #2.