1. 9/8/2015Math SL1 - Santowski1 Lesson 30 - Summation Notation & Infinite Geometric Series Math SL1 - Santowski

2. (A) Review 1. Define a “sequence” 2. Define a “series” 3. State 2 formulas associated with arithmetic sequences and series 4. State 2 formulas associated with geometric sequences and series 9/8/2015 Math SL1 - Santowski 2

3. 9/8/2015 Math SL1 - Santowski 3 (A) Review A sequence is a set of ordered terms, possibly related by some pattern One such pattern is called arithmetic because each pair of consecutive terms has a common difference A geometric sequence is one in which the consecutive terms differ by a common ratio The general term of an arithmetic sequence is defined by the formula u n = u 1 + (n - 1)d The general term of an geometric sequence is defined by the formula u n = u 1 x r (n - 1)

4. 9/8/2015 Math SL1 - Santowski 4 (B) Series - Formulas For an arithmetic sequence then the formula for the sum of its terms is: So in general, the formula for the sum of a geometric series is:

5. 9/8/2015 Math SL1 - Santowski 5 (C) Summation Notation Summation notation is a shorthand way of saying take the sum of certain terms of a sequence  the Greek letter sigma, Σ is used to indicate a summation In the expression i represents the index of summation (or term number), and a i represents the general term of the sequence being summed So therefore,

6. 9/8/2015 Math SL1 - Santowski 6 (D) Ex 1 - Summation Notation Ex 1 – List the terms of the series defined as below, then determine the sum

7. 9/8/2015 Math SL1 - Santowski 7 (D) Ex 1 - Summation Notation Ex 1 – Arithmetic Series

8. (E) Examples of Summation Notation Write out the series expansion for the following and then evaluate the sums:

9. (E) Examples of Summation Notation Evaluate the series that are defined in the following summation notations: 9/8/2015 Math SL1 - Santowski 9

10. 9/8/2015 Math SL1 - Santowski 10 (F) Examples of Summation Notation Ex 2 – List the terms of the series defined as below, then determine the sum

11. 9/8/2015 Math SL1 - Santowski 11 (F) Examples of Summation Notation Ex 2 Geometric Series

12. (F) Examples of Summation Notation Given the series S = 200 + 100 + 50 + 25 … (i) Write a summation expression for the series (ii) Determine S 6 (iii) Determine S 15 (iv) Determine S 21 (v) Predict S 1,000,000 9/8/2015 Math SL1 - Santowski 12

13. (G) Infinite Geometric Series The series ½ + ¼ + 1/8 + 1/16 +....… is an example of an infinite geometric series. (a) Determine the sum of this series. (b) Is it possible to find the sum of any infinite geometric sequence? Explain. (c) Under what conditions is it possible to find the sum of an infinite geometric sequence 9/8/2015 Math SL1 - Santowski 13

14. (G) Infinite Geometric Series Show that the sum of n terms of the series 2 + 1 + ½ + ¼ +..... u n is always less than 4, where n is any natural number. Explain what the following notations mean: 9/8/2015 Math SL1 - Santowski 14

15. (H) Examples of Infinite Geometric Series 9/8/2015 Math SL1 - Santowski 15 Find the sum of the following infinite series, if possible. If not, explain why not.

16. 9/8/2015 Math SL1 - Santowski 16 (I) Internet Links Geometric Sequences & Series From West Texas A&MFrom West Texas A&M Arithmetic Sequences & Series From West Texas A&M UFrom West Texas A&M U

17. 9/8/2015 Math SL1 - Santowski 17 (J) Homework HW: Larson Text, S8.2, p574, Q61-74 Larson Text, S8.3, p583, Q45-67odds