1. MTH 253 Calculus (Other Topics) Chapter 11 – Infinite Sequences and Series Section 11.2 – Infinite Series Copyright © 2009 by Ron Wallace, all rights reserved.

2. Sums of the Terms of a Sequence Adding these terms gives … What would be the sum of the terms of ?

3. Infinite Series The sum of the terms of an infinite sequence is called an infinite series. Notation: NOTES: a k is some function of k whose domain is a set of integers. k can start anywhere (0 or 1 is the most common) The following are all the same:

4. Partial Sums of an Infinite Series Sequence of partial sums. ●●●●●● Recursive Definition:

5. Converging/Diverging Series If converges to then the series converges and If the sequence of partial sums diverges, then so does the series (it has no sum). S is not often easy or even possible to determine!

6. Example … Pattern? NOTE: A general expression for s n is usually difficult to determine.

7. Geometric Series Each term is obtained by multiplying the proceeding term by a fixed constant. Example: NOTE: w/ geometric series, k can start with any value (usually 0 or 1).

8. Geometric Series a is the value of the first term r is the “common ratio” r > 0, all terms have the same sign r < 0, terms alternate signs

9. Geometric Series Under what conditions does a geometric series converge? Case 1a: r = 1 Divergent!

10. Geometric Series Under what conditions does a geometric series converge? Case 1b: r = -1 Divergent!

11. Geometric Series Under what conditions does a geometric series converge? Case 2: |r|  1

12. Geometric Series Under what conditions does a geometric series converge? Case 2: |r|  1 Convergent if |r| < 1; Divergent Otherwise

13. Geometric Series Determine the following sums, if they exists …

14. Telescoping Series

15. Telescoping Series - Examples

16. Hint: “Partial Fractions”

17. a k is the k th term s k is the k th partial sum n th-Term Test NOTE: p  q implies that ~q  ~p, but not ~p  ~q or q  p Proof … The nth-Term Test (aka: The Divergence Test):

18. Algebraic Properties of Infinite Series If …… are convergent … … then … … are convergent. & NOTE: p  q does NOT imply that q  p or ~p  ~q.

19. Algebraic Properties of Infinite Series If … then … … are both convergent or both divergent.

20. Algebraic Properties of Infinite Series If … then … … are both convergent or both divergent. That is, a finite number of terms can be added to or removed from a series without affecting its convergence or divergence.

21. Algebraic Properties of Infinite Series Example: “Change of Index” or “Reindexing”