1. Chapter 9.5
ALTERNATING SERIES

2. 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

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

4. 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.

5. 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:

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

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