Sequences

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.