Calculus II (MAT 146) Dr. Day Friday, April 6, 2018 Sequence and Series Convergence (limits!) (11.1, 11.2) Sequence Characteristics (11.1) Series Worth Remembering (11.2) Your First Tests For Convergence! (11.2, 11.3) The Divergence Test (11.2) The Integral Test (11.3) Friday, April 6, 2018 MAT 146

Some Sequence Calculations If an = 2n−1, list the first three terms of the sequence. The first five terms of a sequence bn are 1, 8, 27, 64, and 125. Create a rule for the sequence, assuming this pattern continues. For the sequence cn = (3n−2)/(n+3) : i) List the first four terms. ii) Are the terms of cn getting larger? Getting smaller? Explain. iii) As n grows large, does cn have a limit? If yes, what is it? If no, why not? Repeat (3) for this sequence: Friday, April 6, 2018 MAT 146

Some Sequence Calculations If an = 2n−1, list the first three terms of the sequence: {1,3,5} The first five terms of a sequence bn are 1, 8, 27, 64, and 125. Create a rule for the sequence, assuming this pattern continues. bn = n3 For the sequence cn = (3n−2)/(n+3) : List the first four terms: 1/4 , 4/5 , 7/6 , 10/7 ii) Are the terms of cn getting larger? Getting smaller? Explain. Growing Larger. iii) As n grows large, does cn have a limit? If yes, what is it? If no, why not? 𝐥𝐢𝐦 𝒏→∞ 𝒄 𝒏 =𝟑 Repeat (3) for this sequence: 𝒅 𝒏 = −𝟏 𝒏 𝟏 𝒏 List the first four terms: −𝟏, 𝟏 𝟐 ,− 𝟏 𝟑 , 𝟏 𝟒 ii) Are the terms of cn getting larger? Getting smaller? Explain. Neither: Alternating. iii) As n grows large, does cn have a limit? If yes, what is it? If no, why not? 𝐥𝐢𝐦 𝒏→∞ 𝒅 𝒏 =𝟎 Friday, April 6, 2018 MAT 146

Sequence Characteristics Convergence/Divergence: As we look at more and more terms in the sequence, do those terms have a limit ? Increasing/Decreasing: Are the terms of the sequence growing larger, growing smaller, or neither? A sequence that is strictly increasing or strictly decreasing is called a monotonic sequence. Boundedness: Are there values we can stipulate that describe the upper limit or lower limit of the sequence? Friday, April 6, 2018 MAT 146

What is an Infinite Series? We start with a sequence {an}, n going from 1 to ∞, and define {si} as shown. The {si} are called partial sums. These partial sums themselves form a sequence. An infinite series is the summation of an infinite number of terms of the sequence {an}. Friday, April 6, 2018 MAT 146

What is an Infinite Series? Our goal is to determine whether an infinite series converges or diverges. It must do one or the other. If the sequence of partial sums {si} has a finite limit as n −−> ∞, we say that the infinite series converges. Otherwise, it diverges. Friday, April 6, 2018 MAT 146

Here are two series we’ve looked at that each converged: Series Convergence Here are two series we’ve looked at that each converged: 𝑛=0 ∞ 𝑥 𝑛 =1+𝑥+ 𝑥 2 + 𝑥 3 +… and 40+10+ 5 2 + 5 8 +… Friday, April 6, 2018 MAT 146

Series Convergence 𝑆=40+10+ 5 2 + 5 8 +…= 𝑗=0 ∞ 40 4 𝑗 =53 1 3 What do we mean by convergence? 𝑆=40+10+ 5 2 + 5 8 +…= 𝑗=0 ∞ 40 4 𝑗 =53 1 3 The infinite series equals a real number. We can describe this using partial sums and limits. 𝑺 𝟏 =40 𝑺 𝟐 =40+10=50 𝑺 𝟑 =40+10+ 5 2 =52.5 𝑺 𝟒 =40+10+ 5 2 + 5 8 =53.125 𝑆 𝑛 =40+10+ 5 2 + 5 8 +…+ 40 4 𝑛−1 and lim 𝑛→∞ 𝑆 𝑛 =53 1 3 Friday, April 6, 2018 MAT 146

lim 𝑛→∞ 𝑘=0 𝑛 𝑥 𝑘 = 1 1−𝑥 , −𝟏<𝒙<𝟏 Series Convergence 𝑛=0 ∞ 𝑥 𝑛 =1+𝑥+ 𝑥 2 + 𝑥 3 +… What do we mean by convergence? lim 𝑛→∞ 𝑘=0 𝑛 𝑥 𝑘 = 1 1−𝑥 , −𝟏<𝒙<𝟏 The power series is equivalent to some function on a known interval. Friday, April 6, 2018 MAT 146

Notable Series A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge? Friday, April 6, 2018 MAT 146

Notable Series The harmonic series is the sum of all possible unit fractions. Friday, April 6, 2018 MAT 146

Notable Series A telescoping sum can be compressed into just a few terms. Friday, April 6, 2018 MAT 146

Fact or Fiction? Friday, April 6, 2018 MAT 146

Our first Series Convergence Test Our first Series Convergence Test . . . The nth-Term Test also called The Divergence Test Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146

Friday, April 6, 2018 MAT 146