Summation (Sigma) Calculator — Evaluate Σ Series (Free)

A summation calculator evaluates sigma notation: Σ tells you to substitute each value of the index into the expression and add the results. For example, Σ from k = 1 to 10 of (2k + 1) is 3 + 5 + 7 + … + 21 = 120. Formulas give the same answer faster: 2 · (10 · 11/2) + 10 = 120.

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How to use the summation calculator

1. On the Sigma notation tab, type the expression in k (such as k^2, 1/k^2 or (1/2)^k), the lower limit and the upper limit. Type ∞ or inf as the upper limit for an infinite series.
2. Read the sum as an exact fraction or integer with its decimal value, the number of terms, the first and last terms, the type of series and, for polynomial terms, a closed-form formula in n.
3. Open Show steps to see the power-sum formulas, the geometric series formula or the convergence test used. The chart and table show the running total.
4. Use the Arithmetic series and Geometric series tabs when you know the first term and the common difference or ratio, and Data list for Σx and Σx² in statistics.

What sigma notation means

In Σ from k = 1 to 4 of 3k, the letter k is the index, 1 is the lower limit, 4 is the upper limit and 3k is the expression. Substitute k = 1, 2, 3 and 4 and add: 3 + 6 + 9 + 12 = 30. The number of terms is upper limit − lower limit + 1, so a sum from k = 3 to 10 has 8 terms, not 7.

Summation formulas

Properties that make sums easier

• Constant multiples come out: Σ 2k = 2 Σ k.
• Sums split: Σ (2k + 1) = 2 Σ k + Σ 1 = 2 · 55 + 10 = 120 for k = 1 to 10.
• Change the starting point by subtracting: Σ from k = 3 to 10 equals Σ from 1 to 10 minus Σ from 1 to 2.

Worked examples

Example 1: Σ k² from k = 3 to 10

From 1 to 10 the sum of squares is 10 · 11 · 21/6 = 385. From 1 to 2 it is 1 + 4 = 5. So the sum from 3 to 10 is 385 − 5 = 380.

Example 2: 1 + 2 + 3 + … + 100

This arithmetic series has 100 terms, first term 1 and last term 100, so the sum is 100 · (1 + 100)/2 = 5050. That is Gauss’s pairing trick: 50 pairs that each add to 101.

Example 3: 2 + 6 + 18 + … (6 terms)

Each term is 3 times the one before, so a₁ = 2 and r = 3. The sum is 2(1 − 3⁶)/(1 − 3) = 2 · (−728)/(−2) = 728.

Example 4: Σ (1/2)ᵏ from k = 1 to ∞

The first 20 terms add to about 0.999999. Because |r| = 1/2 is less than 1, the infinite sum is a₁/(1 − r) = 0.5/0.5 = 1.

Infinite series: does the sum converge?

An infinite series has a sum only when its partial sums approach a single number. The calculator checks these tests in order and names the one it used:

For series that are not geometric, the calculator adds many terms and estimates the remaining tail, then reports the decimal to the accuracy it can trust. When the value agrees with a known constant, it says so: Σ 1/k² matches π²/6, Σ (−1)ᵏ⁺¹/k matches ln 2, and Σ 1/k! from k = 0 matches e. A few series do not fit these tests (for example Σ 1/(k ln k)); then you see the partial sums instead of a guess.

Summation in statistics

Statistics formulas are full of Σ. For the data 4, 7 and 9, Σx = 20, so the mean is 20 ÷ 3 ≈ 6.67. Σx² = 16 + 49 + 81 = 146, which is different from (Σx)² = 400. The sum of squared deviations is Σx² − (Σx)²/n = 146 − 400/3 = 38/3 ≈ 12.67, the starting point for variance and standard deviation.

Common mistakes

• Miscounting terms. From k = 3 to 10 there are 10 − 3 + 1 = 8 terms.
• Using a k = 1 formula for a sum that starts elsewhere. Subtract the part you do not want, or shift the index.
• Confusing Σx² with (Σx)². Square first, then add, for Σx².
• Thinking terms that go to 0 always give a finite sum. The terms of Σ 1/k go to 0, yet the sum grows without bound.
• Using a₁/(1 − r) when |r| ≥ 1. The formula only works for |r| less than 1.

Summation and series practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What is sigma summation?

It is a short way to write a long sum. Σ from k = 1 to 4 of 3k means 3 · 1 + 3 · 2 + 3 · 3 + 3 · 4 = 30. The index runs through every whole number from the lower limit to the upper limit.

How do you do summation on a calculator?

On a TI-84, use sum(seq(expression, variable, start, end)). For example, sum(seq(2X+1, X, 1, 10)) returns 120. Many scientific calculators have a Σ( key with the same four inputs. This page gives the same answer with the steps written out.

What is summation in statistics?

Σx means add every data value. It appears in the mean (Σx/n), the variance and the standard deviation. For 4, 7 and 9, Σx = 20 and the mean is about 6.67. The Data list tab computes Σx, Σx², the mean and Σ(x − x̄)².

How do you calculate a sum of squares?

For the first n squares use n(n + 1)(2n + 1)/6. For n = 10 that is 10 · 11 · 21/6 = 385. For a data set, square each value and add, so Σx² for 4, 7, 9 is 146.

Is summation calculus?

Finite sums are algebra and are taught in Algebra 2 and precalculus. Infinite series and the convergence tests belong to calculus (usually Calculus 2), and sums of rectangles lead to the definite integral.

Can a summation calculator find an infinite sum?

Yes, when the series converges. Type ∞ as the upper limit. Geometric series with |r| less than 1 get an exact answer, such as Σ (1/2)ᵏ = 1. Other convergent series get a decimal with the test used, and divergent series such as Σ 1/k are reported as divergent.

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