Sum of Series Calculator – Arithmetic & Geometric

Series come in families, each with a closed form. The arithmetic series 1+2+…+n sums to n(n+1)/2, so the first 100 integers total 100×101/2 = 5,050. Squares follow Σk² = n(n+1)(2n+1)/6: for n = 3 that is 3×4×7/6 = 14 (indeed 1+4+9 = 14). Geometric series a₁ + a₁r + a₁r² + … sum to a₁(rⁿ−1)/(r−1); for a₁ = 3, r = 2, n = 4 the total is 3×15 = 45. When |r| < 1 the infinite sum converges: 1 + ½ + ¼ + … = 1/(1−½) = 2. The calculator identifies which family your series belongs to, applies the matching formula with full substitution, and cross-checks by direct addition for small n. Partial sums are tabulated so the growth pattern (quadratic for Σk², exponential for geometric r > 1) is visible.

1. 1

   Enter the series type and terms

   Arithmetic, geometric, or a raw list of terms

2. 2

   Apply the closed form

   The matching formula substitution is displayed

3. 3

   Cross-check by addition

   Small n values are summed directly as verification

4. 4

   Test convergence

   Geometric series report S∞ when |r| < 1

• 1

  Σk = n(n+1)/2 and Σk² = n(n+1)(2n+1)/6 cover integer series

• 2

  Geometric sums need r ≠ 1; r = 1 means n equal terms

• 3

  Infinite geometric sums exist only for |r| < 1

• Using n(n+1)/2 for squares — Σk² has its own (2n+1)/6 factor

• Dividing by (1−r) when r > 1 for a finite sum; that form needs (r−1)

• Summing a divergent series like 1+2+4+… to a finite "answer"

• Off-by-one: summing to n−1 after miscounting terms