Calculus

Lesson 52

Alternating Series

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            So far, we've only looked at series where each term was +ve, except for

                                    simple geometric series.

                        An example of an alternating series:

                        Alternating series test:

                                      converges if             1)  for all n  (take derivative!!!!)

                                                                        and      2)                  note

                        Example:     converges even though the harmonic doesn't.

                        Alternating series remainder:                

                                                                                    IF S exists.  Proof on page 583, but it's really

                                                                                    common sense from the conditions above!

            Example:   Approximate S using 6 terms

                        Use the alternating series test to check...it does converge.

                        Using the first 6 terms,        

                                    So

            Absolute vs. Conditional Convergence

                        (NOT JUST FOR PURE ALTERNATING SERIES!!!!)

                        if  converges, then    

                                                                        (note the converse is NOT true!  ex/harmonic)

                        if  converges, then  is absolutely convergent

                        if  converges but  is divergent,

                                                                        then  is conditionally convergent.

                        Examples:        abs.                          div.

                                                  cond.                       cond.

                        Note that rearrangement doesn't work for alternating series!!!!!!

On to Lesson 53 - Ratio and Root Tests

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