Sigma Notation and Series
November 10, 2015
Sigma Notation and Series
Remember from last class that:
A sequence . . .
• Is a list of terms
• Can be specified by a rule/equation
• Can be infinite or finite
• Is a function whose input is consecutive integers and the output are the terms
The following are sequences . . .
A: 2, 4, 6, 8, 10, . . .
B: 1, 4, 9, 16, 25
C: -5, 10, -15, 20, -25, . . .
D: 3, 9, 15, 21 . . .
E:
What is a series?
The following are series . . .
A: 2 + 4 + 6 + 8 + 10
B: 25 + 50 + 75 + . . . + 250
C: 1 + 4 + 7 + 10 + 13 + 16 + 19 + . . .
D:
A series . . .
• Is when the terms of a sequence are added together
• Can be infinite or finite
• Which of the series above are finite?
• Which are infinite?
IBSL Year 1
Sigma Notation and Series
November 10, 2015
Can you tell the difference? Which one is the series?
*We can represent the sum of values in a form called Sigma Notation*
We write series using summation notation
Example: 2 + 4 + 6 + 8 + 10
upper limit of
summation
Can be written in
summation notation as:
index of
summation
Note: n or k typically can be
used in place of i
(or almost any variable!)
IBSL Year 1
general
term or rule
lower limit of
summation
Sigma Notation and Series
November 10, 2015
What if the series is infinite??
Example: 2 + 4 + 6 + 8 + 10 + . . .
Can be written in
summation notation as:
upper limit of
summation is just
the infinity symbol
What does the following sigma notation represent?
How could you calculate the above sum?
Substitute in all the integer values of n
from the lower limit to the upper limit.
You will include the upper limit of
summation #! Then add together.
IBSL Year 1
Sigma Notation and Series
November 10, 2015
*index of summation may
be n, i, j, k, or x
=
Write in words the correct way to read the statement above.
the sum of -3j + 10 as j goes from 1 to 4.
What is the index of summation? __________
What is the general term? _______________
What is the lower limit of summation? _______
What is the upper limit of summation? _______
What is the sum? _________________________
=
Write in words the correct way to read the statement above.
the sum of 4 plus the quantity i - 1 times 5, as i goes from 3 to 6.
What is the index of summation? __________
What is the general term? _______________
What is the lower limit of summation? _______
What is the upper limit of summation? _______
What is the sum? _________________________
IBSL Year 1
*notice the first term in the sum
does not start with n = 1.
Sigma Notation and Series
November 10, 2015
Evaluate (find the sum of the following series):
80
IBSL Year 1
50
253
Sigma Notation and Series
November 10, 2015
Write the following series using summation notation:
1.
2.
3.
4.
5.
6.
4
Find the sum for the general rule that is in Sigma Notation.
What about for:
100
Finding the sum of 100 terms seems like a lot of work,
can we use our GDC?
IBSL Year 1
Sigma Notation and Series
November 10, 2015
Let's use our GDC to find sums quicker!!!
1. Hit Alpha and then y= button.
Scroll over to the FUNC menu.
(or Alpha -> Window)
2. Choose option 2, for Sigma.
3. Enter in all parts of your Sigma Notation and hit Enter.
4. Answer: is the sum of the sequence from the lower limit to
upper limit of summation.
Let's take the same Sigma from before.
Use your GDC to find the sum.
100
Practice using your calculator to find the sums of the following series:
1.
IBSL Year 1
105
2.
50
Sigma Notation and Series
November 10, 2015
Examples: Use your CALC to find the sum of the following Sequences
18
504
Homework
Chapter 6.4
6F: 1 - 3
Need extra practice?
Haese and Harris
Chapter 2 - E.1:1-3; E.2:1,3; E.3:1,4
IBSL Year 1
11,718