1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 9.4 Series Demana, Waits, Foley, Kennedy

2 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. What you’ll learn about Summation Notation Sums of Arithmetic and Geometric Sequences Infinite Series Convergence of Geometric Series … and why Infinite series are at the heart of integral calculus.

3 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Summation Notation

4 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Try These

5 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. How quickly could you determine the sum of the 1 st 100 counting numbers? Without a calculator! Sum of a Finite Arithmetic Sequence

6 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Karl Friedrich Gauss One of the most famous legends in the lore of mathematics concerns the German mathematician Karl Friedrich Gauss (1777–1855), whose mathematical talent was apparent at a very early age. One version of the story has Gauss, at age ten, being in a class that was challenged by the teacher to add up all the numbers from 1 to 100. While his classmates were still writing down the problem, Gauss walked to the front of the room to present his slate to the teacher. The teacher, certain that Gauss could only be guessing, refused to look at his answer. Gauss simply placed it face down on the teacher’s desk, declared “There it is,” and returned to his seat. Later, after all the slates had been collected, the teacher looked at Gauss’s work, which consisted of a single number: the correct answer. No other student (the legend goes) got it right. The important feature of this legend for mathematicians is how the young Gauss got the answer so quickly

7 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Sum of a Finite Arithmetic Sequence

8 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

9 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Let’s try it: Summing the Terms of an Arithmetic Sequence

10 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

11 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn…

12 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Sum of a Finite Geometric Sequence

13 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example Identify the common ratio. Identify the number of terms. Use the formula!

14 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn

15 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Infinite Series

16 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Sum of an Infinite Geometric Series

17 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Summing Infinite Geometric Series

18 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

19 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn…

20 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Repeating Decimals back to fractions?

21 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn…