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Math & Statisticssequences

Summation (Sigma) Calculator

Summation over a range for the standard expressions, with the closed form shown where one exists. Gauss got the first of these at seven years old and it still saves the loop.

Also called: sigma notation calculator, series sum.

Sum
5,050

The sum of k from 1 to 100 is 5,050, across 100 terms. n(n+1)/2 = 100(101)/2

Number of terms
100
Average term
50.5
What was summed
k
Closed form, where one exists
n(n+1)/2 = 100(101)/2
How this is calculated
Method and background

How this is calculated

Each of these sums has a formula that gives the answer directly. The sum of the first n whole numbers is n times n plus one over two, which is why pairing the ends works. Squares and cubes have their own, and the sum of cubes is famously the square of the sum of the numbers themselves. The reciprocal sum is the harmonic series and has no closed form at all, so it is computed term by term and grows without limit, just extremely slowly.

the common sums have closed forms that avoid looping at all
n
The upper limit

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

the first hundred numbers

Sum of
k (the numbers themselves)
From k =
1
To k =
100

Sum5,050

Gauss's pairing: 100 * 101 / 2

Open this example

the first ten squares

Sum of
k squared
From k =
1
To k =
10

Sum385

10 * 11 * 21 / 6, worked by hand

Open this example

a single term is itself

Sum of
k (the numbers themselves)
From k =
5
To k =
5

Sum5

degenerate case

Open this example

Method and limits

What it assumes

  • Integer limits, with the sum running upwards.

What it deliberately does not model

  • Arbitrary expressions are not parsed. The list here covers the standard cases.
  • The harmonic sum accumulates floating-point error over very long ranges.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Does the harmonic series converge?
No. It grows without bound, but so slowly that reaching a total of twenty needs more than a hundred million terms.