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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.

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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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