11.2 Series

22 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.

33 Example: What do we mean when we express a number as an infinite decimal? For instance, what does it mean to write  = The convention behind our decimal notation is that any number can be written as an infinite sum. Here it means that where the three dots (... ) indicate that the sum continues forever, and the more terms we add, the closer we get to the actual value of .

44 Infinite Series In general, if we try to add the terms of an infinite sequence we get an expression of the form a 1 + a 2 + a a n +... which is called an infinite series (or just a series) and is denoted, for short, by the symbol

55 Partial Sum of an Infinite Series We consider the partial sums s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 s 4 = a 1 + a 2 + a 3 + a 4 and, in general, s n = a 1 + a 2 + a a n = These partial sums form a new sequence {s n }, which may or may not have a limit.

66 Definitions If lim n  s n = s exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series  a n.

77 Series Thus the sum of a series is the limit of the sequence of partial sums. So when we write we mean that by adding sufficiently many terms of the series we can get as close as we like to the number s. Notice that

88 Example: Geometric Series a + ar + ar 2 + ar ar n– = a  0 Each term is obtained from the preceding one by multiplying it by the common ratio r. If r = 1, then s n = a + a a = na  Since lim n  s n doesn’t exist, the geometric series diverges in this case.

99 Geometric series Constant ratio between successive terms. Example: 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 Example: The behavior of the terms depends on the common ratio r.

10 If r  1, we have s n = a + ar + ar ar n-1 and rs n = ar + ar ar n-1 + ar n Subtracting these equations, we get s n – rs n = a – ar n

11 If –1< r < 1, we know that as r n  0 as n , so Thus when | r | < 1 the geometric series is convergent and its sum is a/(1 – r ). If r  –1 or r > 1, the sequence {r n } is divergent and so, by Equation 3, lim n  s n does not exist. Therefore the geometric series diverges in those cases.

12 Summary

13  If r is between −1 and +1, the terms of the series become smaller and smaller, and the series converges to a sum.  If r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. The series diverges.  If r is equal to one, all of the terms of the series are the same. The series diverges.  If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the series has no sum. (example Grandi's series: 1 − − 1 + ···).Grandi's series

14 Series The converse of Theorem 6 is not true in general. If lim n  a n = 0, we cannot conclude that  a n is convergent.

15 Series The Test for Divergence follows from Theorem 6 because, if the series is not divergent, then it is convergent, and so lim n  a n = 0.

16 Series

17 Analogy with functions:

18 Practice:

19 Answers:

20

21 Application of geometric series: Repeating decimals

22 Telescoping series

23 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.