CALTRX

Series Summation Calculator

Find partial sums of arithmetic and geometric sequences. See the complete solution with step-by-step working and formula explanations.

n

Enter your values above to see the results.

Tips & Notes

  • ✓Sum of 1 to n = n(n+1)/2. Sum of 1 to 100 = 100×101/2 = 5050. This is the Gauss formula, discovered by recognizing that pairing first and last terms always gives the same sum: 1+100=101, 2+99=101, and so on for 50 pairs.
  • ✓Sum of squares 1²+2²+...+n² = n(n+1)(2n+1)/6. For n=10: 10×11×21/6 = 385. Verify: 1+4+9+16+25+36+49+64+81+100 = 385.
  • ✓Geometric series S = a(1-rⁿ)/(1-r) for finite n terms. Infinite geometric converges only when |r| < 1: S = a/(1-r). Example: 1+0.5+0.25+... = 1/(1-0.5) = 2. If |r| ≥ 1 the sum diverges.
  • ✓Identify series type before calculating: if each term adds a fixed amount (common difference), it is arithmetic. If each term multiplies by a fixed ratio, it is geometric. Mixing up the formulas gives completely wrong answers.
  • ✓Arithmetic mean of a series equals (first term + last term) / 2. Total sum = mean × number of terms. For the series 3,7,11,15,19: mean=(3+19)/2=11, sum=11×5=55.
  • ✓Sigma notation Σ f(i) from i=a to b means: evaluate f at every integer from a to b and add results. The index variable i is local — it does not affect values outside the sum.

Common Mistakes

  • ✗Applying the geometric series formula when |r| ≥ 1 — the series diverges and there is no finite sum. The formula S = a/(1-r) only applies for infinite series with |r| < 1.
  • ✗Off-by-one error in the term count. The sum from i=1 to n=10 has exactly 10 terms, not 9. When computing arithmetic series, count carefully: last minus first, divided by step, plus 1.
  • ✗Confusing arithmetic and geometric series. Arithmetic: each term adds d (common difference). Geometric: each term multiplies by r (common ratio). Using the wrong formula gives completely wrong results.
  • ✗Using the arithmetic formula for partial geometric sums (or vice versa). Always identify the series type first — check whether terms differ by addition or multiplication.
  • ✗Forgetting to apply the geometric formula correctly when r = 1: the formula S = a(1-rⁿ)/(1-r) fails (division by zero). When r=1, all terms equal the first term, so sum = n × a.
  • ✗Assuming all infinite series converge to a finite value. Most do not. Only special cases (like |r|<1 geometric) have closed-form finite sums. Series like 1+2+3+4+... diverge to infinity.

Series Summation Calculator Overview

A series is the sum of the terms of a sequence. While a sequence lists numbers in order (1, 4, 9, 16, 25...), the corresponding series adds them together (1+4+9+16+25=55). Series appear throughout mathematics, physics, and engineering — from calculating compound interest and loan payments to approximating functions with Taylor polynomials and solving differential equations. This calculator evaluates arithmetic series, geometric series, power series, and custom sigma notation expressions.

Arithmetic Series — sum of a sequence with constant difference d:

Sₙ = n/2 × (first + last) = n/2 × (2a₁ + (n−1)d)
EX: Sum of 1+2+3+...+100 → n=100, a₁=1, last=100 → S₁₀₀ = 100/2×(1+100) = 50×101 = 5,050 (Gauss's formula)
EX: Sum of 3+7+11+15+...to 20 terms → a₁=3, d=4, n=20 → S₂₀ = 20/2×(2×3+19×4) = 10×(6+76) = 820
Geometric Series — sum of a sequence with constant ratio r:
Sₙ = a₁ × (1−rⁿ) / (1−r) for r ≠ 1
EX: 1+2+4+8+16+32 → a₁=1, r=2, n=6 → S₆ = 1×(1−64)/(1−2) = (−63)/(−1) = 63
EX: 3+1+1/3+1/9+1/27 → a₁=3, r=1/3, n=5 → S₅ = 3×(1−(1/3)⁵)/(1−1/3) = 3×(1−1/243)/(2/3) ≈ 4.494
Infinite Geometric Series — converges only when |r| < 1:
S∞ = a₁ / (1−r) when |r| < 1
EX: 1+1/2+1/4+1/8+... → a₁=1, r=1/2 → S∞ = 1/(1−1/2) = 2
EX: 0.999... = 9/10 + 9/100 + 9/1000 + ... → a₁=0.9, r=0.1 → S∞ = 0.9/(1−0.1) = 0.9/0.9 = 1 exactly
Sigma (Σ) notation: Σᵢ₌ₐᵇ f(i) means evaluate f(i) for each integer i from a to b and sum all results.
EX: Σᵢ₌₁⁵ i² = 1+4+9+16+25 = 55 | EX: Σᵢ₌₀⁴ 2ⁱ = 1+2+4+8+16 = 31
Geometric series underlie compound interest, loan amortization, and any system where each period multiplies the previous by a fixed factor. The key insight: if |r| < 1, an infinite geometric series converges to a/1−r. This is why 0.999... = 1 exactly — it is the geometric series 9/10 + 9/100 + 9/1000 + ... = (9/10)/(1 − 1/10) = 1. It is also why loan payments work: the present value of an infinite annuity is a finite number. The three classic sum formulas are worth knowing: Σi = n(n+1)/2 (triangular numbers), Σi² = n(n+1)(2n+1)/6, and Σi³ = [n(n+1)/2]². Gauss reportedly derived the first at age 7 by noticing that 1+100 = 101, 2+99 = 101, ..., giving 50 pairs that sum to 101. The same pairing argument extends to any arithmetic series: pair the first and last terms, multiply by the number of pairs.

Frequently Asked Questions

A series is the sum of the terms of a sequence. The sum of the first n terms is called a partial sum Sₙ. Arithmetic series: Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n−1)d). Example: 1+3+5+7+...+19 (first 10 odd numbers). a₁=1, d=2, n=10. Sₙ = 10/2 × (2×1 + 9×2) = 5 × 20 = 100. Or equivalently: sum of first n odd numbers = n². 10² = 100 ✓.

Geometric series sum: Sₙ = a₁(1−rⁿ)/(1−r) for r ≠ 1. If r=1, Sₙ = n×a₁. Example: 2 + 4 + 8 + 16 + 32 = 2×(1−2⁵)/(1−2) = 2×(−31)/(−1) = 62. Verify: 2+4+8+16+32 = 62 ✓. For |r|<1, the infinite geometric series converges: S∞ = a₁/(1−r). Example: 1 + 1/2 + 1/4 + 1/8 +... = 1/(1−0.5) = 2. Zeno's paradox (crossing half the remaining distance each step) converges to a finite sum.

Sigma notation compresses a sum: Σᵢ₌₁ⁿ f(i) means add f(i) for i from 1 to n. Σᵢ₌₁⁵ i² = 1²+2²+3²+4²+5² = 1+4+9+16+25 = 55. Common formulas: Σᵢ₌₁ⁿ i = n(n+1)/2. Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6. Σᵢ₌₁ⁿ i³ = [n(n+1)/2]². Example: Σᵢ₌₁¹⁰⁰ i = 100×101/2 = 5050. These closed-form formulas replace thousands of individual additions with a single calculation.

Telescoping series collapse when consecutive terms cancel. Σᵢ₌₁ⁿ [1/i − 1/(i+1)] = (1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) + ... Most terms cancel, leaving 1 − 1/(n+1). As n→∞, this approaches 1. Another example: Σᵢ₌₁ⁿ [(i+1)² − i²] = (n+1)² − 1. This equals Σᵢ₌₁ⁿ (2i+1) = 2Σi + n = n(n+1) + n = n²+2n. So (n+1)²−1 = n²+2n ✓.

An infinite series converges if its partial sums approach a finite limit. Tests include: ratio test (|aₙ₊₁/aₙ| → L: converges if L<1, diverges if L>1), root test, integral test, comparison test. The harmonic series 1+1/2+1/3+1/4+... diverges (sum → ∞) despite terms approaching 0. The Basel series 1+1/4+1/9+1/16+... = π²/6 ≈ 1.6449 (converges). Convergence is not guaranteed just because terms get smaller.

The sum 1+2+3+4+... diverges to infinity in standard calculus. However, Ramanujan summation (a regularization technique) assigns it the value −1/12. This result appears in physics — specifically in the Casimir effect and string theory — where renormalization gives physical meaning to formally divergent sums. The value −1/12 is obtained by analytically continuing the Riemann zeta function: ζ(s) = Σn⁻ˢ converges for Re(s)>1, and ζ(−1) = −1/12 by analytic continuation, not by summing the divergent series directly.