Chapter 9.5 ALTERNATING SERIES

After you finish your HOMEWORK you will be able to… Use the Alternating Series Test to determine whether an infinite series converges Use the Alternating Series Remainder to approximate the sum of an alternating series Rearrange an infinite series to obtain a different sum

THEOREM 9.14 ALTERNATING SERIES TEST Condition: Let The alternating series and converge if the following two conditions are met. 1. 2. *The 2nd condition can be modified to require only that

THEOREM 9.14 ALTERNATING SERIES REMAINDER If a convergent alternating series satisfies the condition then the absolute value of the remainder involved in approximating the sum by is less than or equal to the first neglected term.

APPROXIMATING THE SUM OF AN ALTERNATING SERIES For this example we will approximate the sum of the following alternating series by its first 6 terms Convergence by the AST (Thm. 9.14) since:

THEOREM 9.16 ABSOLUTE CONVERGENCE If the series converges, then the series also converges. Be careful…the converse statement is not true!!!

DEFINITIONS OF ABSOLUTE AND CONDITIONAL CONVERGENCE 1. is absolutely convergent if converges. 2. is conditionally convergent if converges but diverges.