Warm-up Write an equation whose limit at infinity does not exist Write an equation whose limit at infinity is zero Write and equation whose limit at infinity is -3.

Review of sums of finite series What is the 50th odd number, counting the number 1 as term number 1? What is the sum of the first 50 odd numbers? A population of mice is doubling each year. If the population starts at 100 mice, is the population after 10 years 𝑡 10 𝑜𝑟 𝑆 10 ? Find the population. Your savings account earns 8% annual interest compounded annually. Is the balance after 18 years 𝑡 18 𝑜𝑟 𝑆 18 ? Find the balance.

Watch this video infinite series animated video https://www.youtube.com/watch?v=jktaz0ZautY The link if you are looking at the pdf file

13.5 Sums of Infinite Series

13.5 Sums of Infinite Series

13.5 Sums of Infinite Series

13.5 Sums of Infinite Series

Classwork Find the sum of each series: a. 1+ 1 3 + 1 9 + 1 27 +… b. 1 2 − 1 4 + 1 8 − 1 16 +… c. 1+.1+.01+.001+… d. 1+3+9+27+… 2. Express 0.3333… as an infinite geometric series. Find the sum.

Classwork = 1 1− 1 3 = 1 2 3 S= 𝑡 1 1−𝑟 = 𝟑 𝟐 a. = 1 2 1− − 1 2 Find the sum of each series: a. 1+ 1 3 + 1 9 + 1 27 +… b. 1 2 − 1 4 + 1 8 − 1 16 +… = 1 1− 1 3 = 1 2 3 S= 𝑡 1 1−𝑟 = 𝟑 𝟐 a. = 1 2 1− − 1 2 = 1 2 3 2 S= 𝑡 1 1−𝑟 = 𝟏 𝟑 b.

Classwork S= 𝑡 1 1−𝑟 = 1 1−.1 = 1 .9 = 𝟏𝟎 𝟗 c. d. 𝑟=3, Diverges Find the sum of each series: c. 1+.1+.01+.001+… d. 1+3+9+27+… S= 𝑡 1 1−𝑟 = 1 1−.1 = 1 .9 = 𝟏𝟎 𝟗 c. d. 𝑟=3, Diverges

Classwork .3+.03+.003+.0003+… S= 𝑡 1 1−𝑟 = .3 1−.1 = .3 .9 = 𝟏 𝟑 2. Express 0.3333… as an infinite geometric series. Find the sum. .3+.03+.003+.0003+… S= 𝑡 1 1−𝑟 = .3 1−.1 = .3 .9 = 𝟏 𝟑

13.6 Sigma Notation

SUMMATION NOTATION This number tells us when to stop (the last integer to sub in). This is the formula to sub into This sign means to sum up each of the terms in the sequence This number tells us what integer to start subbing in to create the terms in a sequence Often we want to sum the terms in a sequence so summation notation is a short-hand way express this.

Summation Notation Ex: Evaluate the following sum: Sol:

13.6 Sigma Notation

13.6 Sigma Notation Try It: Calculator

Classwork Write in expanded form. 1. 2. Express the series in sigma notation. 3. 4+9+16+25+36 4. 1 2 + 2 3 + 3 4 + 4 5 5. 3+6+9+…+300 6. 1 3 + 1 9 + 1 27 + 1 81 +… Evaluate. 7. 8.

Classwork Write in expanded form. 1. 2. = 𝟏 𝟐 + 𝟏 𝟒 + 𝟏 𝟔 + 𝟏 𝟖 1. 2. =2 −1 2 +3 −1 3 +4 −1 4 +5 −1 5 +… =𝟐−𝟑+𝟒−𝟓+…

Classwork Express the series in sigma notation. 3. 4+9+16+25+36 4. 1 2 + 2 3 + 3 4 + 4 5 5. 3+6+9+…+300 6. 1 3 + 1 9 + 1 27 + 1 81 +…

=𝟐𝟎𝟕 =𝟏 Classwork 𝑠𝑢𝑚 𝑠𝑒𝑞 5𝑥−32,𝑥,7,15,1 𝑠𝑢𝑚 𝑠𝑒𝑞 𝑠𝑖𝑛 𝑥∙𝜋÷2 ,𝑥,1,50,1 Evaluate. 7. 8. =𝟐𝟎𝟕 7. 𝑠𝑢𝑚 𝑠𝑒𝑞 5𝑥−32,𝑥,7,15,1 =𝟏 8. 𝑠𝑢𝑚 𝑠𝑒𝑞 𝑠𝑖𝑛 𝑥∙𝜋÷2 ,𝑥,1,50,1

Page 502 #1-9 odds, #23-27 odds Page 508 #1-19 odds Homework Page 502 #1-9 odds, #23-27 odds Page 508 #1-19 odds