Solved Examples: Writing and Evaluating Series with Sigma Notation

📚 Quick Study Guide

• 🔢 Sigma Notation: A concise way to represent the sum of a sequence of terms. The general form is $\sum_{i=m}^{n} a_i$, where $i$ is the index, $m$ is the lower limit, $n$ is the upper limit, and $a_i$ is the expression to be summed.
• ➕ Expanding Sigma Notation: To expand a series in sigma notation, substitute each integer value from the lower limit to the upper limit into the expression and add the results. For example, $\sum_{i=1}^{3} i^2 = 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14$.
• 📐 Evaluating Constant Series: If the expression inside the sigma is a constant $c$, then $\sum_{i=m}^{n} c = c(n - m + 1)$.
• 💡 Evaluating Linear Series: The sum of the first $n$ integers is given by $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.
• 🧮 Evaluating Series of Squares: The sum of the squares of the first $n$ integers is given by $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$.
• 🧪 Properties of Sigma Notation:
  • ∑(ai + bi) = ∑ai + ∑bi
  • ∑c⋅ai = c⋅∑ai

Practice Quiz

1. Question 1: Evaluate the series $\sum_{i=1}^{4} (2i + 1)$.
   1. 5
   2. 10
   3. 20
   4. 24
2. Question 2: Write the series $3 + 6 + 9 + 12$ using sigma notation.
   1. $\sum_{i=1}^{4} i$
   2. $\sum_{i=1}^{4} 3i$
   3. $\sum_{i=0}^{3} 3i$
   4. $\sum_{i=1}^{3} 4i$
3. Question 3: Evaluate the series $\sum_{k=2}^{5} k^2$.
   1. 30
   2. 54
   3. 55
   4. 25
4. Question 4: Evaluate the series $\sum_{i=1}^{3} (i^2 - 1)$.
   1. 8
   2. 9
   3. 10
   4. 12
5. Question 5: Write the series $1 + 4 + 9 + 16 + 25$ using sigma notation.
   1. $\sum_{i=1}^{5} i$
   2. $\sum_{i=1}^{5} i^2$
   3. $\sum_{i=0}^{4} i^2$
   4. $\sum_{i=1}^{5} 2i$
6. Question 6: Evaluate the series $\sum_{j=0}^{4} 2$.
   1. 2
   2. 4
   3. 8
   4. 10
7. Question 7: Evaluate the series $\sum_{n=1}^{5} (3n - 2)$.
   1. 25
   2. 30
   3. 35
   4. 40