Advanced Sequences & Series Calculator

Calculate arithmetic, geometric, Fibonacci sequences and infinite series

Enter values to see the formula
Enter values to see the formula
F_n = F_(n-1) + F_(n-2), where F_1 = 1, F_2 = 1
Enter sequence terms to analyze pattern
Select a series type to see the formula

About Sequences and Series

Sequences are ordered lists of numbers, while series are the sums of sequence terms. They're fundamental in mathematics, physics, and engineering.

Arithmetic Sequences

a_n = a_1 + (n-1)d
Sum = n/2 × (2a_1 + (n-1)d)

Each term is found by adding a constant difference to the previous term.

Example: 2, 5, 8, 11, 14, ... (d = 3)

Geometric Sequences

a_n = a_1 × r^(n-1)
Sum = a_1(1-r^n)/(1-r) if r ≠ 1

Each term is found by multiplying the previous term by a constant ratio.

Example: 2, 6, 18, 54, 162, ... (r = 3)

Fibonacci Sequence

F_n = F_(n-1) + F_(n-2)
F_1 = 1, F_2 = 1

Each term is the sum of the two preceding terms.

Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Golden Ratio: lim(F_n/F_(n-1)) = φ ≈ 1.618

Infinite Series

The sum of all terms in an infinite sequence.

  • Convergent: Sum approaches a finite value
  • Divergent: Sum grows without bound
  • Geometric Series: Converges if |r| < 1
  • Harmonic Series: Diverges (Σ1/n = ∞)

Common Convergence Tests

  • Ratio Test: lim|a_(n+1)/a_n| < 1
  • Root Test: lim∜|a_n| < 1
  • Integral Test: Compare with integral
  • Comparison Test: Compare with known series
  • Alternating Series Test: For alternating series

Applications

  • Finance: Compound interest, annuities
  • Physics: Wave functions, vibrations
  • Computer Science: Algorithms, fractals
  • Biology: Population growth models
  • Engineering: Signal processing, control systems