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Consider the finite arithmetic sequence 2, 4, 6, 8, 10.
Now, consider adding these terms together (taking the sum): 2 + 4 + 6 + 8 + 10.
Such a sequence summation is called a series, and is designated by Sn where n represents the number of terms of the sequence being added.
S5 = 2 + 4 + 6 + 8 + 10
This course will be dealing with finite series: sums of a specified number of terms (not infinite sums).
Sn is often called an nth partial sum,
since it can represent the sum of a certain "part" (portion) of a sequence.
A partial sum customarily starts with a1 and ends with an, adding n terms. |
Partial Sums:
S1 = 2
S2 = 2 + 4
S3 = 2 + 4 + 6
S4 = 2 + 4 + 6 + 8
S5 = 2 + 4 + 6 + 8 + 10
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S1 = a1
S2 = a1 + a2
S3 = a1 + a2 + a3
S4 = a1 + a2 + a3 + a4
S5 = a1 + a2 + a3 + a4 + a5
Sn = a1 + a2 + a3 + a4 + a5 + . . . + an
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The summation of a specified number of terms of a sequence
(a series) can also be represented in a compact form, called summation notation, or sigma notation.
The Greek capital letter sigma , , is used to indicate a sum.
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To write the terms of the series, replace n by the consecutive integers
from 1 to 5, as shown above.
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Problem: |
Solution: |
Evaluate:
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Replace j in the expression (j2 + 1) with the values 1, 2, 3 and 4:
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Notice that the expression (j2 + 1) is placed in a set of parentheses behind the sigma. Without the parentheses, only the j2 would be part of the sigma, with the + 1 added on "after" the sigma was completed. |
Evaluate:
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Notice that the starting value is i = 2. While the starting value is usually 1, it can actually be any integer value. Also notice how ONLY the variable i is replaced with the values 2, 3, and 4:
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Evaluate:
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This is an important pattern strategy to remember!
Notice how raising (-1) to a power affected by the signs of the terms in the series.
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Evaluate:
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Yes, it is possible to calculate a summation on an expression starting with a negative number. Substitute -2, -1, 0 and 1. Remember, however, that when working with sequences, the lowest starting value is 1.
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Evaluate:
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OK, so this is a sneaky one. You know that ln( ex) = x, so this summation is the same as which equals 1 + 2 + 3 + 4 + 5 = 15.
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6. Use sigma notation to represent
3 + 6 + 9 + 12 + ...
for the first 36 terms. |
Look for a pattern based upon the position of each term. Often making a table will let the pattern to be more easily seen.
Sequence formula:
an = 3n |
position |
term |
1 |
3 |
2 |
6 |
3 |
9 |
4 |
12 |
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Possible answer:
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7. Use sigma notation to represent
-2 + 4 - 6 + 8 - 10 + ...
for the first 100 terms. |
Again, look for a pattern. Remember what we saw in example #3 regarding using powers of (-1) to affect the signs of the terms.
Sequence formula:
an = (-1)n•2n |
position |
term |
1 |
-2 |
2 |
4 |
3 |
-6 |
4 |
8 |
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Possible answer:
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8. Cameron is starting a 6 week jogging program. He will jog 8 miles the first week and increase the distance by 10% per week. Using sigma notation, write an expression to represent the total number of miles he will have jogged over the 6 week program. |
An increase of 10%, is equivalent to 110% per week in the number of miles.
Week 1: 8 miles
Week 2: 8 + .10(8) or 1.10(8) miles
Week 3: 1.10(1.10)(8) = (1.10)2(8)
Week 4: 1.10(1.10)(1.10)
(8) = (1.10)3(8)
(and so on ...) The pattern is
(1.10)n-1(8).
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Did you notice that the variable used in the summation symbol (sigma) can be
any letter of your choosing. The sum will be the same, irregardless of the variable used.
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Strategies to remember when trying to find an expression for a sequence (series):
Series |
Possible notation
(partial sum) |
Strategy to remember |
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or  |
Always remember that there is more than one possible answer. |
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Patterns can be either increasing or decreasing. |
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Look to see if a value is being consistently added (or subtracted).
Arithmetic Sequence
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Look to see if a value is being consistently multiplied (or divided).
Geometric Sequence |
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Look to see if the values are "famous" numbers such as perfect squares. |
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Look to see if the signs alternate. Alternating signs can be handled using powers of -1. |
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Partial Sums Create New Sequences:
The sums (answers) from partial sums of a sequence may form an interesting new sequence. Take a look at the partial sums of the summation of positive odd integers:
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Partial Sums:
S1 = 1 = 1
S2 = 1 + 3 = 4
S3 = 1 + 3 + 5 = 9
S4 = 1 + 3 + 5 + 7 = 16
S5 = 1 + 3 + 5 + 7 + 9 = 25 |
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The answers from the partial sums create a sequence of perfect squares.
1, 4, 9, 16, 25 |
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For calculator
help with
Summation
Notation (Sigma)
Click here.
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