Math
Summation Calculator
Evaluate sigma sums in closed form. Compute sum of i, i^2, i^3, harmonic, geometric, and arithmetic series, or sum any preset index expression.
Summation (Sigma Notation)
Sum
5,050
100 terms added.
Formula
Sum of the first n positive integers.
First 6 terms
1, 2, 3, 4, 5, 6, ...
Frequently Asked Questions about the Summation Calculator
What does sigma notation mean?
The Greek letter sigma marks a sum. The expression below sigma sets the start of the index, the value above sigma is the end, and the expression to the right is the term. So sum from i=1 to n of i means add up i for every integer i from 1 through n.
What is the closed form for the sum of the first n integers?
The sum 1 + 2 + ... + n equals n(n+1)/2. This is the Gauss formula. For n = 100 the sum is 5,050. The calculator uses this closed form, so it stays fast even for very large n.
Why does the harmonic sum 1 + 1/2 + 1/3 + ... have no closed form?
The harmonic numbers H_n grow like the natural log of n, but no finite algebraic formula gives the exact value. The calculator adds the terms directly. For n = 4 the sum is 25/12, about 2.0833.
When does a geometric series have a clean closed form?
Whenever the common ratio r is not equal to 1, the sum of a*r^(i-1) from i = 1 to n equals a times (1 - r^n) divided by (1 - r). If r equals 1, every term is a and the sum is n*a. The calculator picks the right branch for you.
Why does the calculator reject some large geometric inputs?
JavaScript stores numbers in 64-bit floating point, so r^n overflows to infinity once n is large enough (around n = 1024 for r = 2). When that happens, no reliable sum exists at standard precision, so the calculator returns no result instead of a meaningless number.