Summation Calculator: Sigma Notation
This summation calculator evaluates sigma notation. Type the expression, choose the index variable, and set the lower and upper limits, or use ∞ for an infinite series. For finite sums it adds every term exactly, so the sum of n from 1 to 100 is 5,050 and the sum of n² from 1 to 10 is 385. For infinite series it adds up to a million terms, estimates the limit, and recognizes famous results such as the sum of 1/n² = ฯ²/6, the alternating harmonic series = ln 2 and the sum of 1/n! = e. It also tells you when a series diverges, for example when the terms do not shrink to zero.
Summation Calculator
Add up a sigma-notation sum, finite or infinite, with partial sums and exact values.
How to use the Summation Calculator
- Type the expression, such as 1/n^2 or 3k+1.
- Choose the index variable.
- Enter the lower and upper limits (∞ allowed).
- Read the sum, exact form and partial sums.
Summation formulas
ฮฃ k (k = 1 to n) = n(n + 1)/2 · ฮฃ k² = n(n + 1)(2n + 1)/6 · ฮฃ arโฟ (n = 0 to ∞) = a/(1 − r) for |r| < 1
Features
- Finite sums up to 5 million terms
- Infinite series with convergence check
- Recognizes ฯ²/6, ln 2, e and fractions
- Table of the first 10 terms and partial sums
- Factorials, powers and alternating signs
Frequently asked questions
What does sigma notation mean?
ฮฃ means add. ฮฃ f(n) from n = 1 to 5 means f(1) + f(2) + f(3) + f(4) + f(5). The number below ฮฃ is the starting value and the number above is the last value.
What is the sum of the first 100 numbers?
The sum of n from 1 to 100 is 100 × 101 ÷ 2 = 5,050, using the formula n(n + 1)/2.
What is the sum of 1/n² from 1 to infinity?
It converges to ฯ²/6, about 1.644934. This result is known as the Basel problem.
How do I know if an infinite series converges?
If the terms do not approach 0, the series diverges. If they do, the partial sums must level off. The harmonic series 1/n has terms that go to 0 but still diverges.
What is the formula for a geometric series?
For |r| < 1, the sum of a·rโฟ from n = 0 to ∞ is a/(1 − r). For example, the sum of 0.5โฟ is 1/(1 − 0.5) = 2.