Summation

Tags

math

Refers to adding up a collection of elements.

Commonly we use the \( \Sigma \) symbol to mean the sum of. The value at the head of it specifies how long this summation will last (for example 20 iterations), the bottom value assigns a variable to reference the current value of the iteration and perhaps initialise the first value (for example \( n = 5 \)), and the body of the sigma is a function that's repeated for each iteration and later summed together. For example:

\begin{align*} \sum_^{20} 2n &= 2(1) + 2(2) + 2(3) + \dots 2(20) \\

              &= 2 + 4 + 6 + \dots 40 \\\\
              &= 420

\end{align*}

We know

\begin{align} \sum_^{n} r^2 &= \frac{1}{6}n(n+1)(2n+1) \
\sum_^{n} r^3 &= \frac{1}{4} n^2 (n+1)^2 \
\end{align}

Note: Summation can be split up, and multiplied. For example

\begin{align*} \sum_^{n} r(r-2) &= \sum_^{n} r^2 - 2r \\

                 &= \sum\_{r=1}^{n} r^2 - 2\sum\_{r=1}^{n} r \\\\
                 &= \frac{n}{6} (n+1) (2n+1) - \frac{2}{2} n (n+1) \\\\
                 &= \dots

\end{align*}