Summation Calculator

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.

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.

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.

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.

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.