AcademyReal Numbers
Academy
Summation Notation
Level 1 - Math I (Physics) topic page in Real Numbers.
Sigma Notation (\(\Sigma\))
Sigma notation provides a compact way to represent the sum of a sequence of terms. The Greek letter \(\Sigma\) (sigma) means "sum".
General Form
Sigma Notation
\[\sum_{i=m}^{n} a_i = a_m + a_{m+1} + \cdots + a_n\]
where:
- \(i\) is the index of summation
- \(m\) is the lower bound (starting value)
- \(n\) is the upper bound (ending value)
- \(a_i\) is the term being summed
Example
Sigma Example
\[\sum_{i=1}^{5} i^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55\]
Properties of Summation
Constant Multiple
\[c \sum_{i=1}^{n} a_i = \sum_{i=1}^{n} c a_i\]
Sum of Sums
\[\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i\]
Constant Sum
\[\sum_{i=1}^{n} c = nc\]
Common Sums
Sum of First n Integers
Arithmetic Series
\[\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\]
Sum of Squares
Sum of Squares
\[\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\]
Sum of Cubes
Sum of Cubes
\[\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2\]
Geometric Series
Geometric Series
\[\sum_{i=0}^{n} ar^i = a \frac{1 - r^{n+1}}{1-r} \quad \text{for } r \neq 1\]
Infinite Geometric Series
For \(|r| < 1\):
Infinite Geometric
\[\sum_{i=0}^{\infty} ar^i = \frac{a}{1-r}\]
Double Summation
When summing over two indices:
Double Sum
\[\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} = \sum_{j=1}^{n} \sum_{i=1}^{m} a_{ij}\]
The order of summation can be interchanged (under appropriate conditions).