Summation Calculator
The tool below evaluates a summation written in sigma notation. Enter the expression to the right of the sigma, with the lower and upper bounds, and the calculator walks through the work step by step.
Type naturally or click ⌨ in the box to use the math keyboard.
For sigma notation use \sum_{i=1}^{n} (or click the Σ button).
Examples: \sum_{i=1}^{10} i, \sum_{k=1}^{n} k^2.
A summation in sigma notation, \(\sum_{i=a}^{b} f(i)\), means add up the values of \(f(i)\) for every integer \(i\) from \(a\) to \(b\). For example, \(\sum_{i=1}^{4} i = 1+2+3+4 = 10\). For sums with variable bounds or infinitely many terms, closed-form formulas like \(\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}\) handle the work.
Worked Examples
Three examples worked by hand — the same logic the calculator uses, just spelled out so you can follow it on paper.
Example 1: Sum of integers — evaluate \(\sum_{i=1}^{20} i\)
Use the closed form \(\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}\) with \(n = 20\):
\[\sum_{i=1}^{20} i = \dfrac{20 \cdot 21}{2} = \dfrac{420}{2} = 210\]
This is the formula young Gauss famously rediscovered to add 1 to 100 in seconds.
Example 2: Sum of squares — evaluate \(\sum_{k=1}^{n} k^2\)
Use the closed form \(\sum_{k=1}^{n} k^2 = \dfrac{n(n+1)(2n+1)}{6}\). For example, with \(n = 10\):
\[\sum_{k=1}^{10} k^2 = \dfrac{10 \cdot 11 \cdot 21}{6} = \dfrac{2310}{6} = 385\]
You can confirm by adding \(1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385\). ✓
Example 3: Infinite geometric series — evaluate \(\sum_{i=0}^{\infty} \dfrac{1}{3^i}\)
This is geometric with first term \(a = 1\) and common ratio \(r = \dfrac{1}{3}\). Since \(\lvert r \rvert < 1\), the series converges. The closed form is \(\dfrac{a}{1 - r}\):
\[\sum_{i=0}^{\infty} \dfrac{1}{3^i} = \dfrac{1}{1 - \tfrac{1}{3}} = \dfrac{1}{\tfrac{2}{3}} = \dfrac{3}{2}\]
If \(\lvert r \rvert \geq 1\), the geometric series diverges — the terms don't shrink fast enough to add to a finite value.
What is a Summation?
A summation is a compact way to add up a list of terms that follow a pattern. The capital Greek letter sigma (Σ) means "sum the following." The expression underneath gives the index variable and where the sum starts; the value on top gives where the sum ends.
For example, \(\sum_{i=1}^{4} i = 1 + 2 + 3 + 4 = 10\). The index \(i\) takes every integer value from 1 up to 4, and you add up all the results.
For finite sums with small bounds, you can just add the terms one by one. For sums with a variable upper bound (like \(n\)) or with infinitely many terms, the calculator uses standard formulas — sum of integers, sum of squares, geometric series, and so on.
Try These Examples
\sum_{i=1}^{10} i— sum of the first 10 integers\sum_{k=1}^{n} k^2— sum of squares\sum_{i=0}^{5} 2^i— finite geometric series\sum_{n=1}^{\infty} \frac{1}{2^n}— infinite geometric series\sum_{k=1}^{100} (2k + 3)— linear sum
Common Summation Formulas
A quick reference for the formulas this calculator uses:
| Sum | Closed form |
|---|---|
| \(\sum_{i=1}^{n} i\) | \(\dfrac{n(n+1)}{2}\) |
| \(\sum_{i=1}^{n} i^2\) | \(\dfrac{n(n+1)(2n+1)}{6}\) |
| \(\sum_{i=1}^{n} i^3\) | \(\left[\dfrac{n(n+1)}{2}\right]^2\) |
| \(\sum_{i=0}^{n} ar^i\) | \(\dfrac{a(1 - r^{n+1})}{1 - r}\) |
| \(\sum_{i=0}^{\infty} ar^i\) | \(\dfrac{a}{1 - r}\) for \(\lvert r\rvert < 1\) |
| \(\sum_{i=1}^{n} c\) (constant) | \(cn\) |
The linearity rules also apply: constants pull out of a summation, and the sum of a sum is the sum of the sums.
Tips for Using the Calculator
- For sigma notation, type
\sumand then use_for the lower bound and^for the upper bound:\sum_{i=1}^{n} - For infinity, type
\infty - The virtual math keyboard has a Σ button for one-click sigma entry
- For powers, use
^:k^2means \(k^2\) - For fractions, use
\frac{a}{b}or justa/b
Frequently Asked Questions
What does sigma notation mean?
Sigma notation is shorthand for adding a list of terms that follow a pattern. \(\sum_{i=1}^{n} f(i)\) means "let \(i\) take every integer value from 1 to \(n\), evaluate \(f(i)\) at each, and add the results." The letter \(i\) is the index variable — you can use any letter (\(k\), \(j\), \(n\), …).
What's the difference between a sequence and a series?
A sequence is a list of numbers, like \(1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \ldots\). A series is the sum of a sequence's terms — \(1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots\). The summation sigma notation always describes a series.
How do I know if an infinite series converges?
For a geometric series \(\sum ar^i\), it converges when \(\lvert r \rvert < 1\). For other series, there are several convergence tests (ratio test, comparison test, integral test). A basic necessary condition: the terms must approach zero. If the terms don't shrink to zero, the series can't converge.
Can I split a summation into smaller summations?
Yes — summations are linear. \(\sum (a_i + b_i) = \sum a_i + \sum b_i\), and you can pull out constant factors: \(\sum c \cdot a_i = c \sum a_i\). That's often the first move when evaluating a complicated summation by hand.
What if my summation starts at 0 instead of 1?
Just adjust the formula. For \(\sum_{i=0}^{n} f(i)\), evaluate \(f(0)\) separately and add it to \(\sum_{i=1}^{n} f(i)\). For example, \(\sum_{i=0}^{n} i = 0 + \sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}\). The closed forms in the reference table assume \(i\) starts at 1, but you can shift bounds as needed.