Calculus II (MAT 146) Dr. Day Wednesday, April 4, 2018 Sequences and Series (Ch 11) Sequence and Series Convergence (limits!) (11.1, 11.2) Sequence Characteristics (11.1) Series Worth Remembering (11.2) Your First Test For Convergence! (11.2) Wednesday, April 4, 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: Wednesday, April 4, 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. iii) As n grows large, does cn have a limit? If yes, what is it? If no, why not? Repeat (3) for this sequence: Wednesday, April 4, 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? Wednesday, April 4, 2018 MAT 146

Compressing Sums We can re-express a sum using a mathematical symbol designed to help us compress the expression. We can it sigma notation. 1+2+4+8+… First, determine the pattern or rule (the function) that describes how to get each term. Here, for instance, if we use k to represent the position of each term in the sum—the FIRST term (k = 1), the SECOND term (k = 2), and so on—each term in the sum can be found by raising 2 to a power related to the position value (k): 1= 2 0 ,2= 2 1 ,4= 2 2 ,8= 2 3 ,… Wednesday, April 4, 2018 MAT 146

Compressing Sums 1= 2 0 ,2= 2 1 ,4= 2 2 ,8= 2 3 ,… 1+2+4+8+… 1= 2 0 ,2= 2 1 ,4= 2 2 ,8= 2 3 ,… position of first term (k = 1) exponent is 0: 0 = k − 1 fourth term (position, k = 4) has exponent 3: 3 = k − 1 So the function describing each term in the sum is 2 𝑘−1 . Wednesday, April 4, 2018 MAT 146

Compressing Sums 1+2+4+8+…= 𝑘=1 ∞ 2 𝑘−1 Now we use the Sigma Notation. The function describing each term in the sum is 2 𝑘−1 . Now we use the Sigma Notation. The ceiling of the summation indicates the last value to use for k, when k is incremented 1 unit at a time. Here, the infinity symbol indicates there is no last value! The function describing each term in the sum. 1+2+4+8+…= 𝑘=1 ∞ 2 𝑘−1 The upper-case Greek letter sigma indicates a sum. The floor of the summation, indicates the symbol used as the input variable (k) and the starting value of that input variable. Wednesday, April 4, 2018 MAT 146

Compressing Sums 1+2+4+8+…= 𝑛=0 ∞ 2 𝑛 1+2+4+8+…= 𝑘=1 ∞ 2 𝑘−1 Here’s another correct compression using the function 2 𝑛 . What has changed? 1+2+4+8+…= 𝑛=0 ∞ 2 𝑛 Wednesday, April 4, 2018 MAT 146

Compress each series using Sigma Notation. (1) 1+3𝑥+9 𝑥 2 +27 𝑥 3 +⋯ (6) 6+𝑘+ 𝑘 2 6 (2) 4+𝑛+ 𝑛 2 4 + 𝑛 3 16 +… (7) 1−𝑥+ 𝑥 2 − 𝑥 3 + 𝑥 4 −⋯ (3) 2+7+12+17+… (8) 1+3+9+27 (4) 40+10+ 5 2 + 5 8 +… (9) 2+1+ 1 2 + 1 4 +… (5) 1+ 1 2 + 1 3 + 1 4 + 1 5 +… (10) 4+𝑛+ 𝑛 2 4 + 𝑛 3 16 +… Wednesday, April 4, 2018 MAT 146

Expand each series using these Sigma Notation representations. (II) (III) (IV) (V) Wednesday, April 4, 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}. Wednesday, April 4, 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. Wednesday, April 4, 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 +… Wednesday, April 4, 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 Wednesday, April 4, 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. Wednesday, April 4, 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? Wednesday, April 4, 2018 MAT 146

Notable Series The harmonic series is the sum of all possible unit fractions. Wednesday, April 4, 2018 MAT 146

Notable Series A telescoping sum can be compressed into just a few terms. Wednesday, April 4, 2018 MAT 146

Fact or Fiction? Wednesday, April 4, 2018 MAT 146

Our first Series Convergence Test Our first Series Convergence Test . . . The nth-Term Test also called The Divergence Test Wednesday, April 4, 2018 MAT 146

Wednesday, April 4, 2018 MAT 146

Wednesday, April 4, 2018 MAT 146