adam.schneider
adam.schneider 2d ago • 0 views

Solved Examples: Writing and Evaluating Series with Sigma Notation

Hey there! 👋 Sigma notation can seem tricky, but with practice, you'll be writing and evaluating series like a pro! Let's dive into some examples and then test your knowledge with a quick quiz. Good luck! 👍
🧮 Mathematics

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james.anderson Jan 2, 2026

📚 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
Click to see Answers
  1. Question 1: C
  2. Question 2: B
  3. Question 3: C
  4. Question 4: B
  5. Question 5: B
  6. Question 6: D
  7. Question 7: A

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