Summation Calculator
Type the expression after the sigma, set the lower and upper limits, and get the sum — exactly, wherever it can be. Polynomial terms such as k or k² are summed by a closed formula that works over any range up to 10¹⁵ terms, geometric terms such as (1/2)^k are summed to infinity when they converge, and everything else is added term by term with a table of the terms. Free, no sign-up.
Use one variable, any letter. Write products with * — k*(k+1) — and powers with ^. Functions: sqrt, abs, ln, log10, exp, sin, cos, tan; constants: pi, e.
A whole number, or ∞ for an infinite series.
Results
385
Sum
10
Terms added
Closed form
Method
Exact
Accuracy
Σ (k = 1 to 10) of k^2 = 385
Closed form
The summand is a polynomial in k, so the running total has a closed form.
f(k) = k²
F(n) = Σ (k = 1 to n) of f(k) = (1/3)n³ + (1/2)n² + (1/6)n
F(10) − F(0) = 385 − 0 = 385
Terms
| Index | Term |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
How to Use the Summation Calculator
- 1. Write the summand. Type f(k) using one variable — any letter except e, which is the constant. Multiply with a star,
k*(k+1), because a name followed by a bracket is read as a function. Use^for powers, and sqrt, abs, ln, log10, exp, sin, cos, tan, pi and e if you need them. - 2. Set the limits. Both are whole numbers and may be negative. The upper limit can be ∞ for an infinite series.
- 3. Read the sum. The cards show the sum, the number of terms, the method used, and whether the answer is exact or a rounded decimal.
- 4. Follow the working. For polynomial and geometric summands the closed formula is written out with your numbers in it, and a table lists the first terms and the last few.
Sigma Notation
The Greek capital sigma means 'add up'. In Σ f(k) from k = a to b, the index k starts at the lower limit a, is raised by 1 each time, and stops at the upper limit b; the terms f(a), f(a + 1), …, f(b) are added. If b is below a there are no terms and the sum is 0 — an empty sum.
Closed Forms for Standard Sums
| Sum | Closed form |
|---|---|
| Σ 1, k = 1 to n | n |
| Σ k, k = 1 to n | n(n + 1) ÷ 2 |
| Σ k², k = 1 to n | n(n + 1)(2n + 1) ÷ 6 |
| Σ k³, k = 1 to n | (n(n + 1) ÷ 2)², the square of Σ k |
| Σ (2k − 1), k = 1 to n | n² — the sum of the first n odd numbers |
| Σ rᵏ, k = 0 to n | (1 − rⁿ⁺¹) ÷ (1 − r), for r ≠ 1 |
| Σ rᵏ, k = 0 to ∞ | 1 ÷ (1 − r), when |r| < 1 |
The calculator does not look these up from a table. For a polynomial summand of degree up to 8 it finds the polynomial from the first twelve values using finite differences, builds the running total F(n) = Σ f(k) for k = 1 to n — also a polynomial, one degree higher — and evaluates F(b) − F(a − 1) with whole-number arithmetic. That works for negative limits too.
Worked Examples
- Gauss's sum. For f(k) = k from 1 to 100 the running total is F(n) = (1/2)n² + (1/2)n, so the sum is F(100) − F(0) = 5050 − 0 = 5050.
- Squares. For k² from 1 to 10, F(n) = (1/3)n³ + (1/2)n² + (1/6)n, and F(10) − F(0) = 385.
- Powers of two. For 2^k from 0 to 10 the terms are geometric with c = 1 and r = 2, so the sum is (1 − 2¹¹) ÷ (1 − 2) = 2047.
- An infinite series. For (1/2)^k from 0 to ∞, |r| = 1/2 < 1, so the sum is 1 ÷ (1 − 1/2) = 2. Starting at k = 1 the sum is 1.
- A telescoping sum. For 1/(k*(k+1)) from 1 to 99 there is no closed form to find, so the terms are added exactly: the result is 99/100 — each term is 1/k − 1/(k + 1), and nearly everything cancels.
Exact where possible, honest where not
Terms built from + − × ÷, whole-number powers, abs and square roots of perfect squares are kept as exact fractions. A term with sin, ln, π or a fractional power is added as a decimal, and the result is then labelled Approximate. Sums of more than 10,000 terms are added term by term only when a closed form exists; otherwise the tool says so instead of running a long loop.
Frequently Asked Questions
- What does the sigma symbol mean?
- Σ is the Greek capital letter sigma and stands for 'sum'. The numbers below and above it are the first and last values of the index, and the expression to its right is evaluated and added for each value. Σ k from k = 1 to 4 means 1 + 2 + 3 + 4 = 10.
- What is the formula for the sum of the first n squares?
- Σ k² from k = 1 to n equals n(n + 1)(2n + 1) ÷ 6. For n = 10 that is 10 × 11 × 21 ÷ 6 = 385. Enter k^2 with limits 1 and n and the calculator also shows the polynomial form (1/3)n³ + (1/2)n² + (1/6)n.
- How do I write the summand?
- Use a single variable and ordinary operators. Write k^2 for k squared, 3k + 2 for 3k + 2, and 2^k for 2 to the power k. Multiplication next to a bracket needs a star: k(k+1). The index variable can be any letter you like, such as n or i.
- When does an infinite series have a finite sum?
- When the terms shrink fast enough. A geometric series c·r^k converges exactly when |r| < 1, and then its sum is its first term divided by 1 − r. The harmonic series 1/k, by contrast, diverges even though its terms shrink. The calculator sums geometric series to infinity and tells you when a series diverges.
- Can I start the index at a number other than 1?
- Yes: the lower limit can be 0, a larger number, or negative. Σ k² from −3 to 3 is 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28. For polynomial summands the closed form handles any limits, with the sum equal to F(b) − F(a − 1).
- Why does it say the answer is approximate?
- At least one term used a function such as sin or ln, the constant π or e, or a fractional power, none of which has an exact fraction form. Those terms are added as decimals, so the final digits are rounded. Sums made only of rational terms stay exact.
- What is a telescoping sum?
- A sum in which consecutive terms cancel, leaving only the first and last pieces. 1/(k(k + 1)) = 1/k − 1/(k + 1), so Σ from 1 to n collapses to 1 − 1/(n + 1) = n/(n + 1), which is 99/100 for n = 99.