Home › Calculators › Summation Calculator

Summation Calculator

Type an expression in n, such as n^2, 2n−1 or 1/n!, and choose the lower and upper limits. The calculator adds every term (up to a million), shows the expansion, closed forms for well-known sums, a partial sums table and a chart of how the sum grows.

Expression in n

Use + − * / ^ ( ) and !, n (or k, i), pi, e and functions sqrt, cbrt, abs, ln, log, exp, sin, cos, tan, floor, ceil, round. 2n and n(n+1) work too.

Lower limit (start n =)

Upper limit (end)

Up to 1,000,000 terms

Decimal places

Partial sums table

Enter your values to see the result.

How to use the Summation Calculator

Type the expression using n as the variable, e.g. n^2 or 1/(n(n+1)).
Set the lower and upper limits of the sum.
Press Calculate (the sum also updates as you type).
Read the total, the number of terms and the expanded steps.
Look at the partial sums table and chart to see whether the series settles down.

How sigma notation works

Σ_n=a^b f(n) = f(a) + f(a + 1) + f(a + 2) + … + f(b). The number of terms is b − a + 1. The variable n is just a counter, so Σ_k=1¹⁰ k² and Σ_n=1¹⁰ n² are the same sum.

Useful rules: a constant factor can move outside the sum (Σ c·f(n) = c·Σ f(n)), sums can be split (Σ (f + g) = Σ f + Σ g), and a constant summed N times is N·c.

Common summation formulas

Sum	Closed form	Example
Σ_n=1^N 1	N	N = 10 → 10
Σ_n=1^N n	N(N + 1)/2	N = 100 → 5,050
Σ_n=1^N n²	N(N + 1)(2N + 1)/6	N = 10 → 385
Σ_n=1^N n³	(N(N + 1)/2)²	N = 10 → 3,025
Σ_n=1^N (2n − 1)	N²	N = 20 → 400
Σ_n=0^N−1 rⁿ	(1 − r^N)/(1 − r)	r = 2, N = 10 → 1,023
Σ_n=1^∞ 1/2ⁿ	1	converges
Σ_n=1^∞ 1/n²	π²/6 ≈ 1.644934	Basel problem
Σ_n=0^∞ 1/n!	e ≈ 2.718282	converges fast
Σ_n=1^∞ 1/n	diverges	harmonic series (≈ ln N + 0.5772)

Expression examples

3n+2 — linear terms (an arithmetic series)
1/(n(n+1)) — a telescoping series; from 1 to N the sum is 1 − 1/(N + 1)
(-1)^(n+1)/n — the alternating harmonic series, which tends to ln 2 ≈ 0.693147
sqrt(n), ln(n), sin(n)^2, n! — functions and factorials

Frequently asked questions

What is sigma notation?

Sigma notation Σ (n = a to b) f(n) means: evaluate f(n) for every whole number n from a to b and add the results. For example Σ (n = 1 to 4) n² = 1 + 4 + 9 + 16 = 30.

Which functions can I use in the expression?

Operators + − * / ^ and % (remainder), the factorial !, brackets, the constants pi and e, and the functions sqrt, cbrt, abs, ln, log (base 10), log2, exp, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, floor, ceil and round. Trigonometric functions use radians. You can use n, k or i as the index.

What is the sum of the first n natural numbers?

1 + 2 + … + n = n(n + 1)/2. So the sum from 1 to 100 is 100 × 101 / 2 = 5,050. The sum of squares is n(n + 1)(2n + 1)/6 and the sum of cubes is (n(n + 1)/2)².

Can it calculate infinite sums?

It adds a finite number of terms (up to one million). For a convergent series, a large upper limit gives a close approximation, and the partial sums chart shows the value levelling off. For known series it also shows the exact limit, for example Σ 1/n² = π²/6 and Σ 1/n! from 0 = e.

Why do I get an error for some terms?

If any term is undefined or infinite, for example 1/n at n = 0 or sqrt(n) for a negative n, the sum cannot be computed. Change the lower limit or the expression so every term is a real number.

How accurate is the result?

Terms are computed in double-precision floating point and added with Kahan compensated summation, which keeps rounding error very small even for a million terms. Integer results above 2^53 (about 9 × 10^15) may lose their last digits.

Is my expression sent anywhere?

No. The expression is parsed and evaluated by a small parser running in your browser. Nothing is uploaded and no code is executed with eval.

Related tools