Sequences and Series
Trigonometry and Advanced Math – Section 11.1
I. Warm-Up: The Famous Gaussian Sum Formula
Compute the following sums. Try to find tricks to make the computation go efficiently.
1.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
2.
1 + 2 + 3 + 4 + … + 18 + 19 + 20
3.
1 + 2 + 3 + … + 49 + 50
4.
1 + 2 + 3 + … + 99 + 100
5.
1 + 2 + 3 +… + 999 + 1000
6.
1 + 2 + 3 +… + n-1 + n
II. Terms of a Sequence
The book's definition of a sequence: a function whose domain is a set of consecutive integers.
A sequence is composed of terms. When we add all of the terms in a sequence, we get a series.
The above sequences were rather simple. We could write a general rule for the ith term of any of them as:
an = n
We can write more interesting sequences in a similar way:
Example
Suppose we have the general form an = 2n. What are terms 0, 1, 2, 3, and 4?
More Examples
Write the first six terms of each sequence. Then sum the terms.
a.
an = 2n + 3
b.
an = (-2)n-1
c.
an = (-1)n
d.
an = (-1)n(2n)
We often want to write a general form for a sequence. This involves some pattern recognition.
Examples
Find the general form for each:
1 1 1 1
a.
, , , ,...
3 9 27 81

b.
1 1
1 1
 , , , ,...
3 9 27 81
c.
2, 6, 12, 20

III. Sigma Notation
We have a concise way of writing summations:
n
Let

i  1  2  ...  n
i 1
Note that the sigma notation has three parts:
 The starting index (here, i = 1)
 The ending index (here, i = n)
 What is summed, an expression in terms of the index (here, i)
Expand and compute the following sums:
10
1.
i
i 1
5
2.
 2i
i 1
5
3.
 2i
i 0
3
4.
2
i 0
1
i
First Homework Assignment: Algebra II Book, Section 11.1, #9-29 odd, 37-49 odd
More on Sequences
Trigonometry and Advanced Math – Section 11.1
Warm-Up
Yesterday, we developed a formula for the sum of the first n integers.
Write the formula using sigma notation.
Then compute the following sums. First express each in sigma notation. Then evaluate, using the formula if possible.
a.
1 + 2 + 3 + … + 14 + 15
b.
51 + 52 + 53 + … + 199 + 200
c.
2 + 4 + 6 + 8 + … + 98 + 100
d.
3 + 6 + 9 + … + 57 + 60
e.
102 + 104 + 106 + … + 198 + 200
More Formulas
Problem: Sum of 1
We want a formula for 1 added to itself n times.
a. Express this sum in sigma notation.
b.
Give a complete formula for this sum. Explain.
c.
Expand
50
 2, then compute the sum using the formula.
i1

Sum of Squares
Another important formula is the sum of the first n squares. Rather than derive it, we simply state it:
n
i
i1
Examples
Compute each sum using the above formula(s).

a. 1, 4, 9, 16, 25, …, 81, 100
b.
49, 64, 81, 100, 121
c.
0, 3, 8, 15, 24, 35
2

n(n 1)(2n 1)
6
More on Sigma Notation and Pi Notation
Homework problems #71 will go further into properties of Sigma notation. It is very important. To deal with generalities,
first consider explicit examples and see what happens in those cases. Do they generalize?
Related to Sigma notation is Pi notation. The only difference is that Pi notation allows us to express products, rather than
sums, concisely.
So…
Definition:
b
 f (i)  f (a)  f (a 1) ...  f (b 1)  f (b)
i a

Examples:
Evaluate:
5
a.
i
i1

7
b.
i
i1

3
c.
 2i
i1

4
d.
e
i
i1
Question: What is another way of writing
n
 i?
i1

Homework: Section 11.1: 51-63 odd, 64-66, 71