How to use the summation notation calculator
- Type the expression in k, such as k^2, 2k - 1, 1/k or (1/2)^k.
- Enter the start value of k and the end value. Type inf as the end for an infinite geometric series.
- Read the sum, the number of terms and the average term. A table shows the first terms with running totals.
- Change the expression or limits to compare sums, or use the example buttons to see common series.
Formula
Sum from k = m to n of f(k) = f(m) + f(m+1) + ... + f(n)
Arithmetic: sum of k from 1 to n = n (n + 1) / 2, and of k^2 = n (n + 1)(2n + 1) / 6
Geometric: sum of a r^k from k = 0 to n = a (1 - r^(n+1)) / (1 - r)
Infinite geometric series with |r| < 1: a / (1 - r) from the first term a
Terms are added with compensated summation, so long sums of decimals stay accurate to about 12 digits.
Classic sums
The sum of k from 1 to 100 is 5050, which matches 100 x 101 / 2. The sum of k^2 from 1 to 10 is 385, the sum of 2^k from k = 0 to 10 is 2047 (one less than 2^11), and the sum of the odd numbers 2k - 1 from 1 to 50 is 2500, which is 50 squared. The sum of k factorial from 1 to 10 is 4,037,913, and the harmonic sum of 1/k to 20 is about 3.59773965714.
For an infinite series, the terms (1/2)^k from k = 0 add up to 2, since a = 1 and r = 1/2 give 1 / (1 - 1/2). The series with first term 3 and ratio -2/3 gives 3 / (1 + 2/3) = 1.8.
How to read sigma notation
The big letter sigma means "add up". Below it is where k starts, above it where it stops, and to the right is the rule for each term. So the sum from k = 1 to 4 of k^2 means 1 + 4 + 9 + 16 = 30. The index letter is only a name; the sum does not depend on it. This calculator always calls it k.
Many sums have closed forms that avoid adding term by term. For sequences defined by a first term and a step or ratio, use the arithmetic and geometric sequence calculator. For series that approximate functions, see the Taylor series calculator.
What the expression may contain
You can use numbers, k, parentheses, the operators plus, minus, times, divide and ^ for powers, k! for factorials of whole numbers up to 170, the constants pi and e, and the functions sqrt, abs, ln, log (base 10), exp, sin, cos, tan, floor and ceil. A number directly next to k or a bracket multiplies, so 2k means 2 times k and k(k+1) means k times k + 1.
If the expression cannot be evaluated at some k, for example 1/k at k = 0, the page tells you which k caused it. Sums can have up to one million terms. For logs and exponents, the log, exponent and root calculator is useful for checking single values.
Assumptions and limits
- The variable is k only. Other letters are rejected with a message.
- Finite sums have at most 1,000,000 terms and start and end at whole numbers from -1,000,000,000 to 1,000,000,000.
- An infinite sum is supported only for a geometric series whose ratio is between -1 and 1.
- Sums are computed in floating point and shown to 12 significant digits.
Frequently asked questions
What is sigma notation?
A short way to write a sum. The sigma sign, the start and end values of k, and a rule for each term say exactly which numbers to add.
Can I sum to infinity?
Yes, for a geometric series with a ratio between -1 and 1. Type inf as the end value. Other infinite series are not supported because their sums cannot be found reliably from a few terms.
Why does the harmonic sum show 3.59773965714?
Adding 1/k for k from 1 to 20 gives 3.5977396571436819, rounded to 12 digits. The harmonic series grows slowly without bound.
Can the terms be negative?
Yes. Expressions such as (-1)^k work, and the running total in the table shows how the sum moves up and down.