Math 214

**Test 3 Review**

1) Convince me that the convergence of a series does not depend upon the first few terms of a series.

2) Know how to read a simple theorem, for example:

If the series _{}converges, then the sequence _{} converges to 0.

This theorem is of the form, If A, then B, where A is the hypothesis, or conditions

to be met, and B is the conclusion.

The **converse** of a theorem is found by reading it
backwards: If B, then A.

The converse is not always true. Here, the converse is

If the sequence _{}converges to 0, then the series _{}converges.

This is not true since the sequence _{} converges to 0, but
the series diverges.

Sometimes the **contrapositive** of a theorem is
useful. This is found by writing

If **not** B, then **not **A.

In the above theorem, the contrapositive results in the useful theorem:

If the sequence _{}does not converge to 0, then the series _{}diverges.

This is the Nth term test for divergence.

3) I will not put a host of problems on the test and ask you
to test for convergence. Rather,

I will tell you which test to use (you will need to know how to use all of

the tests) and I will test you on your understanding of the results.

4) Be able to approximate a function using a Taylor or Maclaurin polynomial. I think

I will give you Taylor’s theorem so you will not have to memorize it.

5) Know what a sequence of partial sums is, and what purpose it serves. I will give

you one for which the general term is easy to find. Page 559 example 1a would be

worth your time.

**That’s all folks!**