12.3. Sigma Notation http://www.ck12.org
Example B
Write the sum in sigma notation: 2+ 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
Solution:
2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =
10
i∑= 2 i
Example C
Write the sum in sigma notation.
112 + 212 + 312 + 412 + 512 + 612 + 712
Solution:
112 + 212 + 312 + 412 + 512 + 612 + 712 =∑^7
i= 1
i^12
Concept Problem Revisited
The hardest part when first using sigma representation is determining how each pattern generalizes to thekthterm.
Once you know thekthterm, you know the argument of the sigma. For the sequence creating the series below,
ak=k^2. Therefore, the argument of the sigma isi^2.
1 + 4 + 9 + 16 + 25 +···+ 144 = 12 + 22 + 32 + 42 +··· 122 =
12
i∑= 1 i^2Vocabulary
Sigma notationis also known assummation notationand is a way to represent a sum of numbers. It is especially
useful when the numbers have a specific pattern or would take too long to write out without abbreviation.
Guided Practice
- Write out all the terms of the sigma notation and then calculate the sum.
4
k∑= 03 k−^1 - Represent the following infinite series in summation notation.
(^12) + (^14) + (^18) + 161 +···
- Is there a way to represent an infinite product? How would you represent the following product?
1 ·sin(^3603 )·sin(^3604 )·sin(^3605 )·sin(^3606 )·sin(^3607 )·...
Answers:
4
k∑= 03 k−^1 = (^3 ·^0 −^1 )+(^3 ·^1 −^1 )+(^3 ·^2 −^1 )+(^3 ·^3 −^1 )+(^3 ·^4 −^1 )
=− 1 + 2 + 5 + 8 + 11
- There are an infinite number of terms in the series so using an infinity symbol in the upper limit of the sigma is
appropriate.
(^12) + (^14) + (^18) + 161 +···= 211 + 212 + 213 + 214 +···=∑∞
i= 1
21 i