Covered today: Basic definitions of Taylor polynomials and Taylor series. Interval of convergence.

A summary of what we've covered on sequences and series

For next time: For each of the following functions, compute its Taylor series about the given value of x. Graphically estimate the interval of convergence of the series, and then use the ratio test to determine the radius of convergence analytically.

1.

f(x) = e^-2x about x=0

2.

$f(x) = \frac{1}{(1-x)^2}$ about x=0

3.

$f(x) = \sqrt{4+x}$ about x=0

4.

$f(x) = \frac{1}{\sqrt{1+3x}}$ about x=0

5.

$f(x) = \ln(2x-3)$ about x=2

Use Taylor series to evaluate the following limits. You may use the fact that Taylor series may be added termwise and formally multiplied within their radius of convergence.

1.

$\lim_{x \rightarrow 0} \frac{x-\sin x}{x^3}$ (this one you can check your answer using L'Hopital)

2.

$\lim_{x \rightarrow 0} \frac{(e^{2x}-1)(\ln(1+x^3))}{(1-\cos 3x)^2}$