CalcNotes Sigma Notation and Limits of Finite Sums

Section 5.2 Sigma Notation and Limits of Finite Sums

Definition 5.2.1. Sigma Notation.

Sigma, or summation, notation is

\begin{equation*} \sum_{k=1}^7 a_k \end{equation*}

where

\(\sum\) is Sigma
\(k\) is the index of summation
\(1\) and \(7\) are the lower limit and upper limit
\(a_k\) is the formula for the \(k\)th term

Example 5.2.2.

Compute \(\displaystyle\sum_{k=1}^4 3k\)

Fact 5.2.3. Algebraic rules for summations.

Sum rule: \(\displaystyle\sum (a_k + b_k) = \sum a_k + \sum b_k\)
Difference rule: \(\displaystyle\sum (a_k - b_k) = \sum a_k - \sum b_k\)
Constant multiple rule: \(\displaystyle\sum ca_k = c \sum a_k\)
Constant value rule: \(\displaystyle\sum_{k=1}^n c = nc\)

Example 5.2.4.

Compute \(\displaystyle\sum_{k=1}^7 (2k+3)\)

Fact 5.2.5. Some Sum Formulas.

\(\displaystyle \displaystyle\sum_{k=1}^n k = \frac{n(n+1)}{2}\)
\(\displaystyle \displaystyle\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}\)
\(\displaystyle \displaystyle\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2\)

Example 5.2.6.

Reconsider the area under \(y=x^2+1\) on the interval \([0,1]\text{.}\)

Recall that we computed the upper sum as

\begin{equation*} \frac{1}{4}f(\frac{1}{4}) +\frac{1}{4}f(\frac{2}{4}) +\frac{1}{4}f(\frac{3}{4}) +\frac{1}{4}f(\frac{4}{4}) = \sum_{k=1}^4 \frac{1}{4}f(\frac{k}{4}) \end{equation*}

Breaking it into more intervals:

\begin{equation*} \sum_{k=1}^n \frac{1}{n} f(\frac{k}{n}) = \ldots = 1+\frac{1}{n^3}\sum_{k=1}^n k^2 = 1+ \frac{n(n+1)(2n+1)}{6n^3} = 1 + \frac{2n^3+3n^2+n}{6n^3} \end{equation*}

As the number of intervals increases:

\begin{equation*} \lim_{n\rightarrow\infty} \sum_{k=1}^n \frac{1}{n} f(\frac{k}{n}) = \lim_{n\rightarrow\infty} 1 + \frac{2n^3+3n^2+n}{6n^3} = 1 + \frac{2}{6} = \frac{4}{3} \end{equation*}

Definition 5.2.7. Riemann Sum.

Suppose \(f\) is a function defined on a closed interval \([a,b]\) and assume \(f\) is bounded. For a given positive integer \(n\text{,}\) pick \(n-1\) points \(x_1, x_2, \ldots, x_{n-1}\) such that \(a \lt x_1 \lt x_2 \lt \cdots \lt x_{n-1} \lt b\text{.}\) Set \(x_0=a\) and \(x_n=b\) so we have \(a=x_0 \lt x_1 \lt x_2 \lt \cdots \lt x_{n-1} \lt x_n=b\text{.}\) The set \(P=\{x_0,x_1,\ldots,x_n\}\) is a partition of \([a,b]\text{.}\) \(P\) divides \([a,b]\) into closed subintervals \([x_0,x_1],[x_1,x_2],\ldots, [x_{n-1},x_n]\text{.}\) The width of the \(k\)th subinterval is denoted \(\Delta x_k = x_k-x_{k-1}\text{.}\) We set \(||P||\text{,}\) the norm of \(P\) equal to the largest \(\Delta x_k\text{.}\) In each \([x_k,x_{k-1}]\text{,}\) pick a \(c_k\text{.}\) Set

\begin{equation*} S_P = \sum_{k=1}^n f(c_k) \Delta x_k \end{equation*}

which is called a Riemann sum for \(f\) on \([a,b]\text{.}\)