Sequences & Series

MATH 109 - PrecalculusS. Rook

Overview

• Section 9.1 in the textbook:– Infinite sequences– Factorial notation– Partial sums & summation notation

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Infinite Sequences

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Infinite Sequences• We have discussed finite (countable) lists of numbers

when constructing a table of values:– Given a function f(x), pick values of x to get f(x)– We do this about 2 or 3 times to get an idea what f(x) looks

like– Represents only a subset of the values of f(x)– i.e. a Finite Sequence

• Infinite Sequence: a function whose domain is the natural numbers. The results that are generated from a sequence are its terms

• There are many infinite sequences of interest to mathematicians and scientists– Prime numbers, Fibonacci numbers, etc.

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Terms of a Sequence

• The nth term of a sequence also called the general term is usually written an = f(n)

• Given a natural number k such that 1 ≤ k ≤ n, we can find the kth term of the sequence by simply substituting– i.e. ak = f(k)

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Alternating Sequences

• Alternating Sequence: a sequence in which subsequent terms change from positive to negative or vice versa – Has a general term such as an = (-1)n + 1 · f(n)

• Substitute as before to evaluate a term

Infinite Sequences (Example)

Ex 1: For each sequence, find the first three terms and then the 10th term:

a)

b)

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43 nan

1

11

n

nb nn

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Recursive Sequences

• Recursive Sequence: a sequence defined in terms of itself using previous terms– Usually given at least the first term of the

sequence– e.g. an + 1 = 5 + an; a1 = 2

Recursive Sequences (Example)

Ex 2: Find the first three terms of the recursive sequence:

a)

b)

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24;2 11 nn aaa

11 1;1 nn bnbb

Factorial Notation

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Factorial Notation

• Suppose we were give the recursive sequence an = n · an – 1; a1 = 1

n = 2: a2 = 2 · a1 = 2 · 1 = 2

n = 3: a3 = 3 · a2 = 3 · (2 · 1) = 3 · 2 = 6

n = 4: a4 = 4 · a3 = 4 · (3 · 2 · 1) = 4 · 6 = 24

: : an = n · (n – 1) · (n – 2) · … · 2 · 1

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Factorial Notation (Continued)

• an = n · (n – 1) · (n – 2) · … · 2 · 1 is used often enough that it is given the special name factorial and written as n!

n! means the product of n down to 13! = 3 · 2! = 3 · 2 · 1! = 3 · 2 · 1 = 6

1! AND 0! are both equivalent to 1n! = n · (n – 1)!

• We can use factorials when performing Algebraic operations– By expanding the factorial into a product

Factorial Notation (Example)

Ex 3: Evaluate the factorials by hand:

a)

b)

c)

13

!4

!6

!2!5

!8

!16

!6

kk

Partial Sums & Summation Notation

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Partial Sums• We have seen how to generate successive terms from the

sequence an = f(n)• Another important series concept is the summation of

these terms• The summation through the nth term is called the nth partial

sum denoted Sn

S1 = a1

S2 = a2 + a1

S3 = a3 + a2 + a1

:Sn = an + an-1 + … + a2 + a1

• Each of the nth partial sums forms a sequence• Sn is also called a finite series

Summation Notation

• A shorthand way to write the partial sum from the mth term to the nth term where m ≤ n is where ∑ means to

sum the elements from m to n of the sequence an

m is known as the lower limit (starting value) of the summation (does not always have to start at 1)n is known as the upper limit (ending value) of the summationk (in this case) is known as the index of summation (other variables can be used as well)

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nmm

n

mkk aaaa

1

Summation Notation (Continued)

– The summation of ALL terms of an infinite sequence is known as an infinite series denoted in summation notation as

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1

21i

ii aaaa

Summation Notation (Example)

Ex 4: Evaluate:

a) b)

c) d)

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4

1i

i

5

1

11k

k

9

6 1

11

j jj

1 10

1

xx

Summary• After studying these slides, you should be able to:

– Calculate the terms of the following types of sequences:• Infinite• Alternating• Recursive

– Understand factorial notation and be able to perform simple calculations– Evaluate partial sums and series of a sequence using summation

notation

• Additional Practice– See the list of suggested problems for 9.1

• Next lesson– Arithmetic Sequences & Partial Sums (Section 9.2)

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