Infinite Series: Geometric Series Concept & Applications

1. 11.2 • Series

2. Sequences and Series • A series is the sum of the terms of a sequence. • Finite sequences and series have defined first and last terms. • Infinite sequences and series continue indefinitely.

3. Infinite Series • In general, if we try to add the terms of an infinite sequence • we get an expression of the form • a1 + a2 + a3 + . . . + an + . . . • which is called an infinite series (or just a series) and is denoted, for short, by the symbol

4. nth Partial Sum of an Infinite Series • We consider the partial sums • s1 = a1 • s2 = a1 + a2 • s3 = a1 + a2 +a3 • s4 = a1 + a2 + a3 + a4 • and, in general, • sn =a1 + a2 + a3 + . . . + an= • These partial sums form a new sequence {sn}, which may or may not have a limit.

5. Definitions • If limnsn = s exists (as a finite number), then we call it the sum of the infinite series: an.

6. Special series: Geometric Series • A geometric Series is the sum of the terms of a Geometric Sequence: • a +ar +ar2 + ar3 + . . . + arn–1 +. . . = a 0 • Each term is obtained from the preceding one by multiplying it by the common ratior.

7. Geometric series - Examples: • Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, and finance. • Common ratio: the ratio of successive terms in the series

8. Geometric series: nth partial sum • If r 1, we have • sn = a +ar + ar2 + . . . + arn-1 • and rsn = ar + ar2 + . . . + arn-1 + arn • Subtracting these equations, we get • sn– rsn = a– arn

9. Geometric series: limit of the nth partial sum • If –1< r < 1, we know that as rn 0 as n , When | r | < 1 the geometric series is convergent and its sum is: a/(1 – r ). • If r  –1 or r > 1, the sequence {rn} is divergent and so • limnsn is infinite. The geometric series diverges in • those cases.

10. Summary: Geometric series • When r=-1, the series is called Grandi's series, and is divergent.

11. All Series: Convergence and Divergence • The converse of this Theorem is not true in general. • If limnan = 0, we cannot conclude that an is convergent!

12. Series: Test for Divergence • The Test for Divergence follows from Theorem 6 because, if the series is not divergent, then it is convergent, and so • limnan = 0.

13. Analogy between series and functions:

14. Practice:

15. Answers:

17. Application of geometric series: Repeating decimals

18. Special case: Telescoping series

19. A term will cancel with a term that is farther down the list. • It’s not always obvious if a series is telescoping or not until you try to get the partial sums and then see if they are in fact telescoping.