Summation calculator

What is summation notation?

Sigma notation Σ is shorthand for adding a series of terms. Σ(k=1 to n) k² means: add k² for k = 1, 2, 3, ..., n. So Σ(k=1 to 4) k² = 1 + 4 + 9 + 16 = 30.

Common summation formulas

Σk (k=1 to n) = n(n+1)/2. Sum of first n integers.

Σk² (k=1 to n) = n(n+1)(2n+1)/6. Sum of first n squares.

Σk³ (k=1 to n) = [n(n+1)/2]². Sum of first n cubes.

Σrᵏ (k=0 to n) = (1−r^(n+1))/(1−r). Geometric series.

Properties of summation

Linearity: Σ(af(k) + bg(k)) = aΣf(k) + bΣg(k). You can factor out constants and split sums.

Shifting index: Σ(k=1 to n) f(k) = Σ(k=0 to n−1) f(k+1). You can shift the starting index.

Telescoping: Σ(f(k+1) − f(k)) = f(n+1) − f(1). Most terms cancel.