Summation Calculator

What is a sum?

A sum is the result of adding numbers together. Because addition is commutative and associative, the order and grouping of the terms do not change the total: 1 + 2 + 3 + 4 equals 4 + 1 + 3 + 2. Negative terms are allowed, and a total that includes them is sometimes called an algebraic sum.

When the terms follow a pattern, such as the first hundred whole numbers or the squares of 1 through 12, mathematicians write the sum compactly with the capital Greek letter sigma (Σ). This calculator handles both situations: an arbitrary list of numbers, and a pattern written in sigma notation.

Simple sum mode

Select Simple sum, paste or type your numbers and press Calculate. Separate them with commas, spaces, tabs or new lines, in any combination. Use a dot for decimals: 1.5 is one and a half, while 1,5 is read as the two numbers 1 and 5.

• Negative numbers keep their minus sign, so “-5, 2, 3” adds to 0.
• Text that is not a number is ignored.
• The total is rounded to four decimal places. Whole totals appear bare, one or two decimals are shown with two places (15.50), and three or four decimals with four (0.3750).
• Large totals use thousands separators: 1000000 and 2500000.75 add to 3,500,000.75.

Sigma notation mode

Sigma notation names the quantity to add, the variable that changes, and where it starts and stops. In the sum below the variable is n, it starts at the lower limit 1 and stops at the upper limit 10, and each value of n is put into the expression 2n + 3.

Σ (n = 1 to 10) of (2n + 3) = 5 + 7 + 9 + … + 23 = 140

Fill in the upper limit, the sigma expression, the lower limit and the variable name. The index moves from the lower limit upward in steps of 1, so a lower limit of 0.5 and an upper limit of 3.5 uses the values 0.5, 1.5, 2.5 and 3.5. The upper limit must not be below the lower limit, and the two may differ by at most 10,000.

The result shows the Sum and the first ten terms of the series, each term printed with up to ten decimals.

Write the expression the way you would on paper:

• Operators: + (add), − (subtract), * (multiply), / (divide), ^ (power), and parentheses for grouping.
• A leading minus works as negation, and implicit multiplication is understood: 2n means 2 × n and n(n+1) means n × (n+1).
• Constants: pi and e.
• Variable: the letter you choose in “Variable name” (n, i, x, y, z, a, b or c). Using any other letter gives an error.

A term that has no real value, such as sqrt(n−3) when n is 1 or 2, is shown as nan, and the sum becomes nan as well. A division by zero stops the calculation with an error message.

Closed-form shortcuts

Many common sums have a formula, so you do not need to add the terms one by one. You can use these to check the calculator.

Σ (k = 1 to n) k = n(n + 1) ÷ 2

Σ (k = 1 to n) k² = n(n + 1)(2n + 1) ÷ 6

Σ (k = 0 to n) r^k = (rⁿ⁺¹ − 1) ÷ (r − 1), for r ≠ 1

• The first 100 whole numbers add to 100 × 101 ÷ 2 = 5,050.
• The squares of 1 to 100 add to 100 × 101 × 201 ÷ 6 = 338,350. Enter n^2 with limits 1 and 100 to see the same figure.
• Powers of two: Σ 2ⁿ from n = 1 to 12 is 2¹³ − 2 = 8,190.
• Powers of three from k = 0 to 9: (3¹⁰ − 1) ÷ 2 = 29,524.
• The sum 1/(n(n+1)) from n = 1 to 9 telescopes to 1 − 1/10 = 0.90.
• The harmonic sum 1/n from 1 to 5 is 137/60, about 2.2833333333; there is no simple closed form for larger ranges.

Two properties help with by-hand work: a constant factor can be pulled out, Σ c·f(n) = c · Σ f(n), and a sum of two expressions can be split, Σ (f + g) = Σ f + Σ g. Adding a constant c to every one of the (upper − lower + 1) terms contributes c × (upper − lower + 1).

Other ways to write a sum

Short sums are often written with an ellipsis, 1 + 2 + … + 99 + 100, or in list form, [1, 2, 3, 4]. Such shorthand is fine when the pattern is obvious but loses precision for anything more complicated, which is why sigma notation spells out the expression and limits. For the sequence behind a series, see the sequence calculator. To turn a total into an average, use the mean calculator, and for unequal importance the weighted average calculator.

Common mistakes

• Counting the terms wrongly. A sum from 1 to 10 has ten terms, but a sum from 0 to 10 has eleven: upper − lower + 1.
• Typing a decimal comma. 3,5 becomes the two numbers 3 and 5.
• Mismatching the variable. If the expression uses x, choose x in “Variable name”.
• Forgetting that a negative base with a fractional exponent has no real value, as in (-1)^(n/2).
• Expecting an infinite series. The calculator adds a finite range only.