1. 13.5 – Sums of Infinite Series Objectives: You should be able to…

2. Formulas  The goal in this section is to find the sum of an infinite geometric series. However, this objective is very closely connected to the limit of an infinite sequence.  Compare the sum of infinite geometric series to that of a finite series.  InfiniteFinite

3. How did we get there?  Consider the following sequence of partial sums:  using the finite geom. formula  simplified

4. Continued..  Now consider the limit of.  Since the sequence of partial sums has a limit of 1, we say that the infinite series has a sum of 1 as well.

5. Limits  If the infinite sequence of partial sums ( ) has:  A finite limit, then it converges to the sum of S  An infinite (approaches infinity or no limit) it is said to diverge.

6. Also noted:  If, the infinite geometric series converges to the sum  If and, then the series diverges.

7. Example:  Find the first three terms of an infinite geometric sequence with sum 16 and common ratio.

8. Example:  Show that the series is geometric and converges to if, where n is an integer.

9. Example:  The infinite, repeating decimal 0.4545454545….. can be written as the infinite series 0.45 + 0.0045 + 0.000045 + ….  What is the sum of this series?

10. Example:  What is the sum of the series for 5.363636… ?

11. INTERVAL OF CONVERGENCE