Sequences

Calculate terms and sums for sequences and series.

Sequences, Series & Convergence Handbook

Mathematical Principles & Theorems

Analyzes infinite sequences \((a_n)\) and series \(\sum_{n=1}^\infty a_n\). (1) Arithmetic Progression: \(a_n = a_1 + (n-1)d\), sum \(S_n = \frac{n}{2}(2a_1 + (n-1)d)\); (2) Geometric Progression: \(a_n = a_1 r^{n-1}\), sum \(S_n = a_1 \frac{1-r^n}{1-r}\), infinite sum \(S_\infty = \frac{a_1}{1-r}\) for \(|r| < 1\); (3) Fibonacci sequence \(F_n = F_{n-1} + F_{n-2}\) with Binet formula \(F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}\); (4) Convergence tests: Ratio Test \(L = \lim |a_{n+1}/a_n|\), Root Test, and Integral Test.

Operating Instructions

Select sequence family: Arithmetic, Geometric, Fibonacci, Harmonic, or Custom \(a_n\).
Configure initial term \(a_1\), common difference \(d\) or ratio \(r\), and term count \(N\).
Click Compute to generate term listings, partial sums, and convergence test metrics.
Inspect the sequence growth plot and partial sum convergence trajectory on the visual stage.

Parameters

First term a₁

Common diff d

n-th term (n)

Sum first N terms

First term a₁

Common ratio r

n-th term (n)

Sum first N terms

Number of terms

Find n-th Fibonacci

Number of terms (max 30)

Terms to sum (k)

aₙ = a₁ + (n-1)d —

Sₙ = n(a₁+aₙ)/2 —

Arithmetic Mean —

Terms

aₙ = a₁ · rⁿ⁻¹ —

Sₙ = a(rⁿ-1)/(r-1) —

S∞ (|r|<1) —

Terms

F(k) —

Sum F(1)..F(n) —

φ (Golden Ratio) —

F(n)/F(n-1) —

Fibonacci

Lucas (2, 1, 3, 4, 7, …)