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. 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 .

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

5. 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 + a an =
These partial sums form a new sequence {sn}, which may or may not have a limit.

6. Definitions
If limn  sn = s exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series
 an.

7. Special case: Geometric Series
a + ar + ar2 + ar ar n– = a  0
Each term is obtained from the preceding one by multiplying it by the common ratio r.
Example:
If r = 1, then sn = a + a a = na 
Since limn  sn doesn’t exist, this geometric series diverges.

8. 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
The behavior of the terms depends on the common ratio r.

9. Geometric series: nth partial sum formula
If r  1, we have
sn = a + ar + ar ar n-1
and rsn = ar + ar ar n-1 + ar n
Subtracting these equations, we get
sn – rsn = a – ar n

10. Geometric series: limit of the nth partial sum
If –1< r < 1, we know that as r n  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 {r n} is divergent and so
limn  sn is infinite. The geometric series diverges in
those cases.

11. Recap: Geometric series
When r=-1, the series is called Grandi's series, and is divergent.

12. Series: Convergence and Divergence
The converse of Theorem 6 is not true in general.
If limn an = 0, we cannot conclude that  an is convergent.

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

14. Series

15. Analogy with functions:

16. Practice:

17. Answers:

19. Application of geometric series: Repeating decimals

20. Telescoping series:

21. 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.