Sequence & Series Suite
Arithmetic and geometric sequences with the n-th term and partial sums, infinite geometric convergence with the |r| < 1 check, and the classic power-sum formulas computed. Covers §5.20–5.25 of the calculus material.
Arithmetic Sequence a, a+d, a+2d, …
Geometric Sequence a, ar, ar², …
Partial sums Sₙ — when |r| < 1 they flatten toward S∞ (red bars mark the last few). FIG. T29-1
Σk, Σk², Σk³ up to n
An infinite geometric series converges exactly when |r| < 1, to S∞ = a/(1 − r): each partial sum Sₙ = a(1 − rⁿ)/(1 − r), and rⁿ → 0 kills the correction term. At |r| ≥ 1 the terms never shrink, so the sum cannot settle.