Series And Sequence Calculator Guide
How the Series and Sequence Calculator Works
A sequence is an ordered list of numbers (like 2, 4, 6, 8, ...). A series is the sum of the terms in a sequence (like 2 + 4 + 6 + 8 + ...). Understanding the pattern is key to finding formulas and sums.
Basic Concepts
- Sequence: A list: a₁, a₂, a₃, a₄, ... (Example: 1, 4, 9, 16, 25, ...)
- Series: A sum: a₁ + a₂ + a₃ + a₄ + ... (Example: 1 + 4 + 9 + 16 + 25 + ...)
Arithmetic Sequences and Series
Arithmetic Sequence
Each term is found by adding a constant difference (d) to the previous term.
nth term: aₙ = a₁ + (n - 1)d
Example: 3, 7, 11, 15, ... (d = 4)
Arithmetic Series Sum
Sₙ = n(a₁ + aₙ) / 2
Or equivalently: Sₙ = n[2a₁ + (n-1)d] / 2
Geometric Sequences and Series
Geometric Sequence
Each term is found by multiplying the previous term by a constant ratio (r).
nth term: aₙ = a₁ × r⁽ⁿ⁻¹⁾
Example: 2, 6, 18, 54, ... (r = 3)
Geometric Series Sum
- Finite (n terms): Sₙ = a₁(1 - rⁿ) / (1 - r), when r ≠ 1
- Infinite (|r| < 1): S = a₁ / (1 - r)
Special Sequences and Convergence
Fibonacci Sequence
Each term is the sum of the two preceding ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
aₙ = aₙ₋₁ + aₙ₋₂
Square Numbers
Perfect squares of natural numbers: 1, 4, 9, 16, 25, 36, 49, ...
aₙ = n²
Convergence and Divergence
- Convergent: An infinite series converges if the sum approaches a finite value. Example: 1 + 1/2 + 1/4 + 1/8 + ... = 2
- Divergent: An infinite series diverges if the sum grows without bound or oscillates. Example: 1 + 2 + 3 + 4 + ... = ∞
Real-World Applications
Finance
Compound interest, loan payments, and annuities use geometric sequences and series.
Computer Science
Algorithm analysis, recursion, and data structure capacity calculations.
Physics
Model wave patterns, decay processes, and harmonic motion using series.
Nature
Fibonacci appears in flower petals, pine cones, and spiral galaxies.