A series is the result of adding the terms of a sequence, such as 2 + 4 + 6 + 8. Series are important because they let us describe repeated patterns, total change, accumulated distance, money growth, and many scientific processes in a compact way. Summation notation uses the Greek letter sigma, Σ, to write long sums clearly without listing every term.
Learning this notation helps students move from arithmetic patterns to algebraic formulas and calculus ideas.
Key Facts
- Sigma notation means add terms: Σ from k = m to n of a_k = a_m + a_(m+1) + ... + a_n.
- The index variable, such as k in Σ a_k, is a placeholder and can be renamed without changing the sum.
- Arithmetic sequence term formula: a_n = a_1 + (n - 1)d.
- Arithmetic series formula: S_n = n(a_1 + a_n)/2 = n[2a_1 + (n - 1)d]/2.
- Geometric sequence term formula: a_n = a_1 r^(n - 1).
- Finite geometric series formula: S_n = a_1(1 - r^n)/(1 - r), for r ≠ 1.
Vocabulary
- Series
- A series is the sum of the terms of a sequence.
- Sequence
- A sequence is an ordered list of numbers that often follows a pattern or rule.
- Summation notation
- Summation notation is a compact way to show that many terms should be added together using the symbol Σ.
- Index
- The index is the variable that counts through the terms in a summation, such as k in Σ a_k.
- Common ratio
- The common ratio is the constant factor multiplied from one term to the next in a geometric sequence.
Common Mistakes to Avoid
- Ignoring the lower and upper limits of Σ is wrong because they tell which terms are included in the sum.
- Treating the index variable as a fixed number is wrong because the index changes step by step through the allowed values.
- Using an arithmetic series formula for a geometric series is wrong because arithmetic series add a constant difference while geometric series multiply by a constant ratio.
- Forgetting parentheses in expressions like Σ(2k + 1) is wrong because summation applies to the entire term rule, not just part of it.
Practice Questions
- 1 Evaluate Σ from k = 1 to 5 of 3k.
- 2 Find the sum of the first 8 terms of the arithmetic sequence 4, 7, 10, 13, ...
- 3 Explain why Σ from k = 1 to 4 of k^2 is not equal to (Σ from k = 1 to 4 of k)^2.