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Section 1.5 Summation Notation and Generalizations

Subsection 1.5.1 Sums

  • Most arithmetic operations, like addition, multiplication are binary operations, taking two operands, i.e. numbers.
  • Adding more than two numbers together: \(a_1 + a_2 + a_3 + a_4\) is the same as repeating addition with two operands on partial sums: \(((a_1 + a_2) + a_3) + a_4\).
  • We group operations between ( and ) that must have precedence - must be computed first.
  • Addition and multiplication are associative, for example: \((a_1 + a_2) + a_3 = a_1 + (a_2 + a_3)\). Subtraction, division, modulo (i.e. remainder) are not associative.
  • Summation notation: \( \sum_{k=1}^4 a_k = a_1 + a_2 + a_3 + a_4\)
  • In general, we write a finite series as: \( a_1+a_2+a_3 +\dots +a_n=\sum_{k=1}^{n} a_k\).

    • The variable \(k\) is referred to as the index, or the index of summation.

    • The expression \(a_k\) is the general term of the series. It defines the numbers that are being added together in the series.

    • The value of \(k\) below the summation symbol is the initial index and the value above the summation symbol is the terminal index.

    • It is understood that the series is a sum of the general terms where the index start with the initial index and increases by one up to and including the terminal index.

  1. \(\displaystyle \sum_{i=1}^4 a_i= a_1+ a_2+a_3+a_4\)

  2. \(\displaystyle \sum_{k=0}^5 b_k=b_0+b_1+b_2+b_3+b_4+b_5\)

  3. \(\displaystyle \sum_{i=-2}^2 c_i=c_{-2}+c_{-1}+c_0+c_1+c_2\)

If the general terms in a series are more specific, the sum can often be simplified. For example,

  1. \(\displaystyle \sum_{i=1}^4 i^2=1^2+2^2+3^2+4^2=30\)

  2. \begin{equation*} \begin{split} \sum_{i=1}^5 (2i-1)&=(2\cdot 1-1)+(2\cdot 2-1)+(2\cdot 3-1)+(2\cdot 4-1)+(2\cdot 5-1)\\ & =1+3+5+7+9\\ & =25\\ \end{split}\text{.} \end{equation*}

Subsection 1.5.2 Generalizations

Summation notation can be generalized to many mathematical operations, for example, \(A_1\cap A_2\cap A_3\cap A_4=\underset{i=1}{\overset{4}{\cap }}A_i\)

Definition 1.5.3. Generalized Set Operations.

Let \(A_1, A_2, \ldots , A_n\) be sets. Then:

  1. \(\displaystyle A_1\cap A_2\cap \cdots \cap A_n=\underset{i=1}{\overset{n}{\cap }}A_i\)

  2. \(\displaystyle A_1\cup A_2\cup \cdots \cup A_n=\underset{i=1}{\overset{n}{\cup }}A_i\)

  3. \(\displaystyle A_1\times A_2\times \cdots \times A_n=\underset{i=1}{\overset{n}{\times }}A_i\)

  4. \(\displaystyle A_1\oplus A_2\oplus \cdots \oplus A_n=\underset{i=1}{\overset{n}{\oplus }}A_i\)

If \(A_1 = \{0, 2, 3\}\text{,}\) \(A_2 = \{1, 2, 3, 6\}\text{,}\) and \(A_3 = \{-1, 0, 3, 9\}\text{,}\) then

\begin{equation*} \underset{i=1}{\overset{3}{\cap }}A_i=A_1\cap A_2\cap A_3=\{3\} \end{equation*}

and

\begin{equation*} \underset{i=1}{\overset{3}{\cup }}A_i=A_1\cup A_2\cup A_3=\{-1,0,1,2,3,6,9\}\text{.} \end{equation*}

With this notation it is quite easy to write lengthy expressions in a fairly compact form. For example, the statement

\begin{equation*} A\cap \left(B_1\cup B_2\cup \cdots \cup B_n\right)= \left(A\cap B_1\right)\cup \left(A\cap B_2\right)\cup \cdots \cup \left(A\cap B_n\right) \end{equation*}

becomes

\begin{equation*} A \cap \left(\underset{i=1}{\overset{n}{\cup }}B_i\right)= \underset{i=1}{\overset{n}{\cup }}\left(A\cap B_i\right)\text{.} \end{equation*}

Exercises 1.5.3 Exercises

1.

Calculate the following series:

  1. \(\displaystyle \sum_{i=1}^3 (2 + 3i)\)

  2. \(\displaystyle \sum_{i=-2}^1 i^2\)

  3. \(\sum_{j=0}^n 2^j\text{ }\) for \(n= 1, 2, 3, 4\)

  4. \(\sum_{k=1}^n (2k-1)\) for \(n = 1, 2, 3, 4\)

Answer.

  1. \(\displaystyle 24\)

  2. \(\displaystyle 6\)

  3. \(\displaystyle 3,7,15,31\)

  4. \(\displaystyle 1,4,9,16\)

2.

Calculate the following series:

  1. \(\sum_{k=1}^3 k^n\) for \(n = 1, 2, 3, 4\)

  2. \(\displaystyle \sum_{i=1}^5 20\)

  3. \(\sum_{j=0}^3 \left(n^j+1\right)\) for \(n = 1, 2, 3,4\)

  4. \(\sum_{k=-n}^n k\) for \(n = 1, 2, 3, 4\)

3.

  1. Express the formula \(\sum_{i=1}^n \frac{1}{i(i+1)}= \frac{n}{n+1}\) without using summation notation.

  2. Verify this formula for \(n=3\text{.}\)

  3. Repeat parts (a) and (b) for \(\sum_{i=1}^n i^3=\frac{n^2(n+1)^2}{4}\)

Answer.

  1. \(\displaystyle \frac{1}{1 (1+1)}+\frac{1}{2 (2+1)}+\frac{1}{3 (3+1)}+\cdots +\frac{1}{n(n+1)}=\frac{n}{n+1}\)

  2. \(\displaystyle \frac{1}{1(2)}+\frac{1}{2(3)}+\frac{1}{3(4)}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}=\frac{3}{4}=\frac{3}{3+1}\)

  3. \(1+2^3+3^3+\cdots +n^3=\left(\frac{1}{4}\right)n^2(n+1)^2\) \(\quad 1+8+27=36 = \left(\frac{1}{4}\right)(3)^2(3+1)^2\)

4.

Verify the following properties for \(n = 3\text{.}\)

  1. \(\displaystyle \sum_{i=1}^n \left(a_i+ b_i\right) =\sum_{i=1}^n a_i +\sum_{i=1}^n b_i\)

  2. \(\displaystyle c\left(\sum_{i=1}^n a_i\right) = \sum_{i=1}^n c a_i\)

5.

Rewrite the following without summation sign for \(n = 3\text{.}\) It is not necessary that you understand or expand the notation \(\left( \begin{array}{c} n \\ k \\ \end{array} \right)\) at this point. \((x + y)^n= \sum_{k=0}^n \left( \begin{array}{c} n \\ k \\ \end{array} \right)x^{n-k}y^k\text{.}\)

Answer.

\((x+y)^3=\left(\text{}_0^3\right)x^3+\left(\text{}_1^3\right)x^{2}y+\left.(_2^3\right)x y^2+\left(\text{}_3^3\right)y^n\)

6.

  1. Draw the Venn diagram for \(\underset{i=1}{\overset{3}{\cap }}A_i\text{.}\)

  2. Express in “expanded format”: \(A\cup (\underset{i=1}{\overset{n}{\cap }}B_i)= \underset{i=1}{\overset{n}{\cap }}(A \cup B_n)\text{.}\)

7.

For any positive integer \(k\text{,}\) let \(A_k = \{x \in \mathbb{Q}:k-1 < x \leq k\}\) and \(B_k = \{x \in \mathbb{Q}: -k < x < k\}\text{.}\) What are the following sets?

  1. \(\displaystyle \underset{i=1}{\overset{5}{\cup }}A_i\)

  2. \(\displaystyle \underset{i=1}{\overset{5}{\cup }}B_i\)

  3. \(\displaystyle \underset{i=1}{\overset{5}{\cap }}A_i\)

  4. \(\displaystyle \underset{i=1}{\overset{5}{\cap }}B_i\)

Answer.

  1. \(\displaystyle \{x\in \mathbb{Q}\mid 0 < x \leq 5\}\)

  2. \(\displaystyle \{x\in \mathbb{Q}\mid -5 < x < 5\}=B_5\)

  3. \(\displaystyle \emptyset\)

  4. \(\displaystyle \{x\in \mathbb{Q}\mid -1 < x < 1\}=B_1\)

8.

For any positive integer \(k\text{,}\) let \(A_k = \{x \in \mathbb{Q}:\text0 < x < 1/k\}\) and \(B _k = \{x \in \mathbb{Q}:\,0 < x < k\}\text{.}\) What are the following sets?

  1. \(\displaystyle \underset{i=1}{\overset{\infty }{\cup }}A_i\)

  2. \(\displaystyle \underset{i=1}{\overset{\infty }{\cup }}B_i\)

  3. \(\displaystyle \underset{i=1}{\overset{\infty }{\cap }}A_i\)

  4. \(\displaystyle \underset{i=1}{\overset{\infty }{\cap }}B_i\)

9.

The symbol \(\Pi\) is used for the product of numbers in the same way that \(\Sigma\) is used for sums. For example, \(\prod _{i=1}^5 x_i=x_1 x_2 x_3 x_4 x_5\text{.}\) Evaluate the following:

  1. \(\displaystyle \prod _{i=1}^3 i^2\)

  2. \(\displaystyle \prod _{i=1}^3 (2i+1)\)

Answer.

  1. \(\displaystyle 36\)

  2. \(\displaystyle 105\)

10.

Evaluate

  1. \(\displaystyle \prod _{k=0}^3 2^k\)

  2. \(\displaystyle \prod _{k=1}^{100} \frac{k}{k+1}\)