Mathematical Series
A mathematical series is the sum of the terms of a sequence, either finite or infinite. For a finite series the sum is always a specific number, while an infinite series converges to a finite limit only if its terms decrease fast enough. Series are foundational in calculus, number theory, physics, and engineering — from computing π to modelling oscillations and signal processing.
Key Formula
S_n = a_1 + a_2 + ... + a_n
LaTeX: S_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n
| Symbol | Meaning | Unit |
|---|---|---|
| Sₙ | Partial sum of the first n terms | dimensionless |
| aₖ | kth term of the sequence | dimensionless |
| n | Number of terms | dimensionless |
Worked Example
Problem
Find the sum of the arithmetic series: 4 + 7 + 10 + 13 + ... for the first 20 terms.
Solution
Step 1: Identify a₁ = 4, d = 3, n = 20. Step 2: Last term: a₂₀ = 4 + 19×3 = 4 + 57 = 61. Step 3: Sum: S₂₀ = (n/2)(a₁ + aₙ) = (20/2)(4 + 61) = 10 × 65 = 650.
Answer
S₂₀ = 650.
Common Mathematical Series and Their Sums
| Series Type | Formula | Sum Formula | Converges? |
|---|---|---|---|
| Arithmetic series | a + (a+d) + (a+2d) + ... | Sₙ = n/2 × (2a + (n−1)d) | Finite only |
| Geometric series | a + ar + ar² + ... | Sₙ = a(1−rⁿ)/(1−r) | Yes, if |r| < 1 |
| Harmonic series | 1 + 1/2 + 1/3 + ... | Diverges | No |
| p-series | 1 + 1/2ᵖ + 1/3ᵖ + ... | Converges for p > 1 | Yes, if p > 1 |
| Telescoping series | (1−1/2)+(1/2−1/3)+... | Collapses to finite value | Yes |
Interactive Tools
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Related Terms
Arithmetic Sequence
An arithmetic sequence (also called an arithmetic progression) is an ordered list of numbers in which the difference between consecutive terms is constant, called the common difference d. The nth term is given by aₙ = a₁ + (n−1)d, where a₁ is the first term. Arithmetic sequences model uniform increments such as salary increases, regular savings, and equally spaced physical measurements.
Geometric Sequence
A geometric sequence (or geometric progression) is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio r. The nth term is given by aₙ = a₁ × r^(n−1). Geometric sequences model exponential growth and decay, compound interest, population doubling, and signal attenuation.
Binomial Theorem
The Binomial Theorem provides a formula for expanding any positive integer power of a binomial (a + b)ⁿ as a sum of terms of the form C(n,k) × aⁿ⁻ᵏ × bᵏ, where C(n,k) are binomial coefficients. It eliminates the need to multiply out the expression repeatedly and reveals the coefficient pattern known as Pascal's triangle. The theorem has applications in probability theory, combinatorics, calculus approximations, and algebraic identities.
From Latin "series" (a row, chain, succession). The rigorous treatment of infinite series began with Newton and Leibniz in the 17th century and was formalised by Cauchy and Abel in the 19th century.