Covered today: wrapup of Taylor series. A look at alternating series, and the error incurred by using the nth partial sum to estimate the sum of the series.

For next time: I won't be taking anything up. You are strongly encouraged to go back over your homework that was due on Friday and make sure you know how to do everything on it. Here are a few additional problems to think about:

1.

Explain why the statement $\ln 2 = 1-1/2+1/3-1/4+\cdots$ is not very useful in terms of providing a practical means of estimating $\ln 2$.

2.

Explain how you could build a program that is capable of estimating the sine of any number to within .0001 only using the first 8 terms of the Taylor series of the sine function. Prove that this can be done.

3.

Let

$$a_n = \frac{n^2+2}{n-3n^2}$$

Find the limit of the sequence a_n and prove, using the official definition of convergence, that a_ converges to that number.

4.

Prove, using the definition, that the sequence

$$a_n = \frac{n}{\ln(n+1)}$$

diverges to $\infty$.