1. Review Write an explicit formula for the following sequences.
5, 7, 9, 11, 13,
1, 3, 9, 27, 81,

2. Answers Write an explicit formula for the following sequences.
5, 7, 9, 11, 13,
an = 2n + 3
2. 1, 3, 9, 27, 81,
an = 1 • 3 (n-1)

3. Sum of numbers 1-100?

4. Series and Summation Notation

5. Series A series is the sum of the terms in a sequence.
= 135
= 135

6. a1•(1 – rn) n ________ Sn = – (a1 + an) 2 Sn = (1 – r) Series Formulas
Arithmetic Series
Geometric Series n 2
a1•(1 – rn)
(1 – r)
________
Sn = – (a1 + an)
Sn =

7. Steps for Evaluating a Series
Determine if the terms in the series are arithmetic or geometric.
Common difference or common ratio?
Substitute the information you are given into the appropriate formula.

8. Example 1
Evaluate a series with the terms 1, 7, 13, 19, 25 for the first 13 terms.

9. Example 2
Find the sum of the first 10 terms of the geometric series with a1 = 6 and r = 2.

10. Example 3
A philanthropist donates $50 to the SPCA. Each year, he pledges to donate 12 dollars more than the previous year. In 8 years, what is the total amount he will have donated?

11. I’m just a fancy way of saying, “Add everything up!”
Summation Notation
Instead of saying: “Find the sum of the series denoted by an = 3n + 2 from the 3rd term to 7th term,” mathematicians made up a symbol to deal with it. Sigma! ∑
I’m just a fancy way of saying, “Add everything up!”

12. Summation Notation
last term
sequence formula
first term

13. Example 4: Evaluating Using Summation Notation

14. Writing Series in Summation Notation
Step 1) Determine the explicit formula. Step 2) Identify the lower and upper limits. Step 3) Write the series in summation notation.

15. Example 5
Use summation notation to write the series for the specified number of terms.
…; n = 6

16. Example 6
Use summation notation to write the series for the specified number of terms.
…; n = 9

17. Example 7
Use summation notation to write the series for the specified number of terms.
…; n = 33

18. Example 8
Use summation notation to write the series for the specified number of terms.
…; n = 12

19. Infinite vs. Finite
The difference between a finite and infinite series is whether or not there is a “…” at the end.
Example,
3, 5, 7, 9….
6, 3, 1.5, .75

20. Convergent and Divergent Series

21. What will happen to the terms of this sequences as it continues forever?
It approaches 0!

22. Convergent Sequence
A sequence is converging if its terms approach 0. *Only applies to geometric sequences.

23. Determine if the following sequences are convergent or divergent:

24. Sum of an Infinite Geometric Series
We can find the sum of this series, even though it goes on forever, because it is convergent.

25. Sum of an Infinite Geometric Series
The following conditions must be true to use this formula:
The series must be geometric
The series must be convergent
The series must be infinite

26. Example 1
Determine whether each infinite geometric series diverges or converges. If it converges, find the sum.
a) …
b) …

27. Example 2
Evaluate each infinite geometric series a) b)

28. Formula Recap
Sequence
Series
Arithmetic
Geometric
Finite
Infinite

29. Given 4, 20, 100, 500, … (a) find the explicit formula
(b) find the 15th term
(c) find the sum of the first 15 numbers