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Lesson 23: Sequences and series

Topic: This lesson covers Chapter 23: Sequences and series

WeBWorK: There are three WeBWorK assignments on today’s material:

Sequences – Introduction

Sequences – Arithmetic

Series – Finite Arithmetic

Question of the day: What is a sequence? What is a series?

Answer of the day: A sequence is just a list of numbers. A series is list of numbers, added up.

Lesson Notes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

Introduction to sequences and series

Today we will introduce two very powerful ideas (they are the building blocks of Calculus) – however, the ideas themselves are not complicated: sequences and series. They both have to do with lists of numbers, rather than individual numbers.

Definition 23.1. A sequence is an ordered list of numbers (we call each number in the list a term in the sequence). We write a sequence in order as follows: a_{1}, a_{2}, a_{3}, a_{4},...
In short we write the above sequence as \left\{a_{n}\right\} or \left\{a_{n}\right\}_{n \geq 1}

Let’s look at examples of some sequences. As you look at this example, keep the following questions in mind:

  • Is there a pattern in the sequence?
  • Can you predict what the next term of the sequence will be?
  • Can you predict what the 10th term of the sequence will be? What about the 20th, or the 100th?
  • Can you find a formula for a_n, the nth term of the sequence?

Example 23.2. Here are some examples of sequences.
a) 4,6,8,10,12,14,16,18, \dots
b) 1,3,9,27,81,243, \dots
c) +5,-5,+5,-5,+5,-5, \dots
d) 1,1,2,3,5,8,13,21,34,55, \dots
e) 5,8,-12,4,5.3,7,-3, \sqrt{2}, 18, \frac{2}{3}, 9, \dots

Example 23.3. Consider the sequence \left\{a_{n}\right\} with a_{n}=4 n+3. Calculate the first five terms of the sequence a_1, a_2, a_3, a_4, and a_5. What is the 200th term of the sequence?

VIDEO: Introduction to sequences – Examples 23.2 and 23.3

Example 23.4. Find the first 6 terms of each sequence.
a) a_{n}=n^{2}
b) a_{n}=\frac{n}{n+1}
c) a_{n}=(-1)^{n}
d) a_{n}=(-1)^{n+1} \cdot 2^{n}

Another way to describe a sequence is by giving a recursive formula for the nth term a_{n} in terms of the lower terms. Here are some examples.

Example 23.6. Find the first 6 terms in the sequence described below.
a) a_{1}=4, and a_{n}=a_{n-1}+5 for n>1
b) a_{1}=3, and a_{n}=2 \cdot a_{n-1} for n>1
c) a_{1}=1, a_{2}=1, and a_{n}=a_{n-1}+a_{n-2} for n>2

VIDEO: Two ways of describing sequences -Examples 23.4 and 23.6

Definition 23.8. A series is a sum of terms in a sequence. We denote the sum of the first k terms in a sequence with the following notation:
\sum_{i=1}^{k} a_{i}=a_{1}+a_{2}+\cdots+a_{k}
The summation symbol \sum comes from the greek letter \Sigma, pronounced “sigma,” which is the greek letter for “S.”

Example 23.9. Find the sum.
a) \sum_{i=1}^{4} a_{i}, for a_{i}=7 i+3
b) \sum_{j=1}^{6} a_{j}, for a_{n}=(-2)^{n}
c) \sum_{k=1}^{5}\left(4+k^{2}\right)

VIDEO: Intro to series – Example 23.9

The arithmetic sequence

Definition 23.10. A sequence \left{a_{n}\right} is called an arithmetic sequence if any two consecutive terms have a common difference d. The arithmetic sequence is determined by d and the first value a_{1}. This can be written recursively as:
a_{n}=a_{n-1}+d \quad \text { for } n \geq 2
Alternatively, we have the general formula for the nth term of the arithmetic sequence
a_{n}=a_{1}+d \cdot(n-1)

Example 23.11. Determine if the sequence is an arithmetic sequence. If so, then find the general formula for a_{n}.
a) 7,13,19,25,31, \ldots
b) 13,9,5,1,-3,-7, \dots
c) 10,13,16,20,23, \dots
d) a_{n}=8 \cdot n+3

VIDEO: Arithmetic sequences – Example 23.11

Example 23.12. Find the general formula of an arithmetic sequence with the given property.
a) d=12, and a_{6}=68
b) a_{1}=-5, and a_{9}=27
c) a_{5}=38, and a_{16}=115

VIDEO: Arithmetic sequences – finding a general formula, Example 23.12

Example 23.13. Find the sum of the first 100 integers, starting from 1 . In other words, we want to find the sum of 1+2+3+\cdots+99+100.

VIDEO: Arithmetic series – intro example 23.13

Observation 23.14 . Let \left\{a_{n}\right\} be an arithmetic sequence, whose n th term is given by the formula a_{n}=a_{1}+d(n-1). Then, the sum a_{1}+a_{2}+\cdots+a_{k-1}+a_{k} is given by adding \left(a_{1}+a_{k}\right) precisely \frac{k}{2} times:
\sum_{i=1}^{k} a_{i}=\frac{k}{2} \cdot\left(a_{1}+a_{k}\right)

Example 23.15. Find the value of the arithmetic series.
a) Find the sum a_{1}+\cdots+a_{60} for the arithmetic sequence a_{n}=2+13(n-1)
b) Determine the value of the sum: \quad \sum_{j=1}^{1001}(5-6 j)
c) Find the sum of the first 35 terms of the sequence
4,3.5,3,2.5,2,1.5, \ldots

VIDEO: Arithmetic series – finding the sum, Example 23.15

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