acalculator

What is the value of my summation?

Type the term and the two limits of a sum written in sigma notation. The calculator adds every term and shows the exact total when it is a fraction.

Your numbers

Use + - * / ^, brackets, sqrt, ln, sin, cos, pi and e.
Sum
385

The sum of n^2 for n = 1 to 10 is 385.

Exact sum
385
Number of terms
10
Terms
1 + 4 + 9 + … + 100
Sum of
n^2 for n = 1 to 10

Sum: 385. The sum of n^2 for n = 1 to 10 is 385.

How to calculate

Adds up a term f(n) for every whole number n from a lower to an upper limit (sigma notation), exactly as a fraction when the term allows, with the terms written out.

Example with the default inputs (Term f(n) n^2, Lower limit 1, Upper limit 10): The sum of n^2 for n = 1 to 10 is 385.

Method: Σ f(n) for n = a to b = f(a) + f(a + 1) + … + f(b): the term is worked out at every whole number from a to b and the values are added.

  • The index takes every whole number from the lower limit to the upper limit, both included; the upper limit must be at least the lower one.
  • A term made of numbers, the index, + − × ÷ and whole-number powers is added up exactly in fractions (0^0 counts as 1); other terms are added in floating point with compensated summation.
  • log and ln are the natural logarithm; angles are in radians.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Term f(n) n^2, Lower limit 1, Upper limit 10 gives Sum 385, Exact sum 385, Number of terms 10, Terms 1 + 4 + 9 + … + 100.Source: Sigma notation as in OpenStax College Algebra 2e, section 9.4: https://openstax.org/books/college-algebra-2e/pages/9-4-series-and-their-notations
  2. Term f(n) n, Lower limit 1, Upper limit 100 gives Sum 5,050, Number of terms 100.
  3. Term f(n) 1/k, Lower limit 1, Upper limit 10 gives Sum 2.928968, Exact sum 7381/2520.
  4. Term f(n) (1/2)^i, Lower limit 0, Upper limit 4 gives Sum 1.9375, Exact sum 31/16, Terms 1 + 1/2 + 1/4 + 1/8 + 1/16.
  5. Term f(n) 2n - 1, Lower limit -2, Upper limit 2 gives Sum -5, Terms (-5) + (-3) + (-1) + 1 + 3.
  6. Term f(n) sqrt(n), Lower limit 1, Upper limit 4 gives Sum 6.146264, Terms 1 + 1.414213562 + 1.732050808 + 2.

How it works

The sum Σ f(n) for n = a to b is f(a) + f(a + 1) + … + f(b): the term f is worked out at every whole number from the lower limit a to the upper limit b, both included, and the values are added. There are b − a + 1 terms. The lower limit is a whole number from −100 to 100, the upper limit a whole number from −100 to 1,000, and the upper limit must be at least the lower limit.

The term. Type it with one index letter, n, i, k, or j (or no letter, for a constant term); numbers, read as the exact decimals typed (0.1 is one tenth); +, - (or −), * (or ×, ·), / (or ÷), and ^ (or **) for powers; brackets ( ), [ ], { }; the constants pi (or π) and e; and the functions sqrt, cbrt, abs, exp, ln and log (both the natural logarithm), log10, sin, cos, tan, asin, acos, atan (also arcsin, arccos, arctan), sinh, cosh, and tanh, with angles in radians. Something next to a letter, a function, or a bracket multiplies it (2n, n(n + 1), 3 sin(n)); a function without brackets takes the next power, a number included (sin n^2 is sin(n²), sin 2 is sin(2)), and a function with brackets takes only what is inside them, so a power after the bracket applies to its value (sin(n)^2 is (sin n)²). A number right after a letter, another number, or a closing bracket needs a sign: n2, 1e2 and (n + 1)2 give a message. A + in front of something is ignored (+n, 2 ++ n), and any kind of closing bracket closes any kind of opening one ((n + 1] is (n + 1)). Powers group from the right, and a minus sign in front applies after a power (-n^2 is −(n²)). A negative number to a fraction power has no real value here.

Exact or decimal. When every term is a fraction, that is, the term uses only numbers (read as the exact decimals typed), the index, + − × ÷, and powers whose exponent is a whole number (0^0 counts as 1), each term is worked out exactly and the terms are added exactly. The exact work has a size limit: a power a^e whose exact value would need more than 50,000 bits (e × (bits of a's numerator + bits of its denominator), where the bits of a whole number m are the digits of |m| in base 2), or a running total past 50,000 bits (numerator and denominator together), sends the whole sum to floating point. Otherwise (a function, pi or e, or a power with a fraction exponent), each term is worked out in 64-bit floating point too, and the terms are added with compensated (Neumaier) summation. In floating point, a term that comes out infinite (a division by 0 or a value past the float range), or not a number because a step on the way was infinite (as 2^1331 − 2^1331 in 2^(n^3) - 2^(n^3) at n = 11), gives no answer, "The term is infinite or too large at n = …"; a step past the float range is fine when the term still comes out finite (1/e^800 is 0 to a float). A term that is not a number for any other reason (the square root or logarithm of a negative number, 0 ÷ 0) gives "The term has no real value at n = …". A term that divides by 0 in exact work gives "The term divides by 0 at n = …", and a sum too large for a 64-bit float gives no answer.

The calculator shows:

  • Sum: the total (the exact total rounded to the nearest 64-bit float), shown to at most 6 decimal places.
  • Exact sum: only when every term is a fraction and the total in lowest terms has at most 40 digits in its numerator and at most 40 in its denominator: a whole number (385) or numerator/denominator (7381/2520), with a plain hyphen for a minus sign.
  • Number of terms: b − a + 1.
  • Terms: every term when there are 5 or fewer, else the first three, …, and the last one, joined by a plus sign with a space on each side. A fraction term is written like the exact sum (1/4); a floating-point term is written with up to 10 significant figures and no thousands separators (1.414213562). A negative term is in brackets: (-5) + (-3).
  • Sum of: the term as typed (with spaces at the ends removed and each run of spaces written as one space), then the words "for", the index letter, "=", the lower limit, "to", and the upper limit, with one space between the parts: n^2 for n = 1 to 10.

Worked examples by hand

Σ n² for n = 1 to 10. 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385. The formula n(n + 1)(2n + 1)/6 gives 10 × 11 × 21 ÷ 6 = 385 too.

Σ n for n = 1 to 100. Pair the first and last terms: 1 + 100 = 101, 2 + 99 = 101, and so on, 50 pairs: 50 × 101 = 5050.

Σ 1/k for k = 1 to 10. Over the common denominator 2520: (2520 + 1260 + 840 + 630 + 504 + 420 + 360 + 315 + 280 + 252)/2520 = 7381/2520 ≈ 2.928968.

Σ (1/2)^i for i = 0 to 4. 1 + 1/2 + 1/4 + 1/8 + 1/16 = 16/16 + 8/16 + 4/16 + 2/16 + 1/16 = 31/16 = 1.9375.

Σ (2n − 1) for n = −2 to 2. (−5) + (−3) + (−1) + 1 + 3 = −5.

Σ √n for n = 1 to 4. 1 + 1.414214 + 1.732051 + 2 ≈ 6.146264 (√2 and √3 are not fractions, so the sum is a decimal).

Other questions people ask

What does sigma notation mean?

Σ (the Greek capital sigma) means "add up". Σ n² for n = 1 to 10 means 1² + 2² + 3² + … + 10². The letter under the sigma is the index; it starts at the lower limit and goes up by 1 until it reaches the upper limit.

How do I calculate a summation by hand?

Write out each term by putting each value of the index into the term, then add them. For Σ (2n − 1) from n = 1 to 4: 1 + 3 + 5 + 7 = 16. For long sums, use a formula, such as 1 + 2 + … + n = n(n + 1)/2.

What are the formulas for common sums?

Σ k for k = 1 to n is n(n + 1)/2. Σ k² is n(n + 1)(2n + 1)/6. Σ k³ is (n(n + 1)/2)². A geometric sum Σ rᵏ for k = 0 to n is (1 − rⁿ⁺¹)/(1 − r) when r is not 1.

Why does the answer show a fraction?

When every term is a fraction (the term uses only numbers, the index, + − × ÷ and whole-number powers), the calculator adds them exactly. The sum of 1/k for k = 1 to 10 is exactly 7381/2520, about 2.928968.

Can the limits be negative?

Yes. The index can start below 0: Σ (2n − 1) for n = −2 to 2 is (−5) + (−3) + (−1) + 1 + 3 = −5. The upper limit must be at least the lower limit, and a term such as 1/n has no value at n = 0.

Which letter should I use for the index?

Use n, i, k, or j, and use only one of them in the term. The letter does not change the sum: Σ i² and Σ n² over the same limits are equal.

Can I sum to infinity?

No. This calculator adds a finite number of terms, with the upper limit up to 1,000. For a geometric series with |r| < 1, the infinite sum is a/(1 − r); the sequence calculator shows it.