Summation calculator
Add up a formula over a range of whole numbers: exact value, the closed form and the partial sums.
Use the variable you chose in the expression: k^2 + 1, 3*2^k, (-1/2)^k, 1/(k(k+1)). Limits are whole numbers; type inf for an infinite series.
What does sigma notation mean?
The capital sigma Σ means “add up”. Under it sits the index and where it starts, above it where it stops, and to its right the formula for each term. Σ from k = 3 to 7 of k² means 3² + 4² + 5² + 6² + 7² = 9 + 16 + 25 + 36 + 49 = 135 (OpenStax Algebra and Trigonometry 2e, section 13.4, Example 1). The index is a dummy variable, so n, k, i, j and m are interchangeable (OpenStax Calculus Volume 1, section 5.1), and the lower limit need not be 1. The number of terms is the upper limit minus the lower limit plus 1; here 7 − 3 + 1 = 5.
What are the formulas for sums of powers?
Calculus Volume 1, section 5.1, gives Σ1 = n, Σk = n(n + 1) ÷ 2, Σk² = n(n + 1)(2n + 1) ÷ 6 and Σk³ = n²(n + 1)² ÷ 4, all from k = 1 to n, together with the rules that a constant factor comes out of a sum and that a sum of terms is the sum of the sums. For higher powers the calculator uses Faulhaber’s formula, built from Bernoulli numbers with exact fractions. Example 5.2 of the book: Σ (i − 3)² for i = 1 to 200 expands into Σi² − 6Σi + 9Σ1 = 2,686,700 − 120,600 + 1,800 = 2,567,900, and Σ (i³ − i²) for i = 1 to 6 is 350. Because the closed form does not add terms one by one, the limits can be huge: the sum of k² for k up to a billion is 333,333,333,833,333,333,500,000,000.
How do arithmetic series work?
When consecutive terms differ by a constant, the sum is the number of terms times the average of the first and last: S = n(a₁ + aₙ) ÷ 2 (OpenStax Algebra and Trigonometry 2e, section 13.4). For Σ (3k − 8) from k = 1 to 12, a₁ = −5 and a₁₂ = 28, so S = 12 × (−5 + 28) ÷ 2 = 138, as in Example 2. The calculator recognizes any expression linear in the index, including ones like 5 − 6k or k/2 + 1/4, and shows the first term, the last term and the count. Summing the same linear expression as a combination of Σk and Σ1 gives the same number, which the calculator uses as an internal check on small ranges.
How do finite geometric series work?
If each term is the previous one times a constant ratio r, the sum of the first n terms is S = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1 (OpenStax Algebra and Trigonometry 2e, section 13.4). For Σ 3·2ᵏ from k = 1 to 6, a₁ = 6, r = 2 and n = 6, so S = 6 × (1 − 64) ÷ (1 − 2) = 378 (Example 4). Type the term as a·r^k with any rational ratio, a negative ratio such as (−1/2)^k, or an irrational one such as e^k; shifts and scalings in the exponent like 2^(k+1) or 4^(k−1) are recognized. When a sum mixes kinds of terms, for example 3k² + 2^k, each part gets its own closed form and the parts are added.
When does an infinite series have a sum?
An infinite geometric series with first term a and ratio r has the sum a ÷ (1 − r) when |r| < 1, and diverges when |r| ≥ 1 (OpenStax Calculus Volume 2, section 5.2). Σ 4374·(−1/3)^(k−1) from k = 1 converges to 6561/2 = 3280.5 (Algebra and Trigonometry 2e, Example 7), and Σ (−3)^(n+1) ÷ 4^(n−1) converges to 36/7 (Calculus Volume 2, Example 5.9). The series with terms e^(2n) has ratio e² ≈ 7.39, so it diverges. A polynomial in the index, such as Σk, always diverges, because its terms do not approach 0. For p-series 1/kᵖ the test is p > 1 converges, p ≤ 1 diverges (Calculus Volume 2, section 5.3): the calculator reports the verdict, for Σ 1/k² it says converges, but does not compute the sum itself.
How do I type the expression and the limits?
Write the formula with the index letter you picked: k^2 + 1, 3*2^k, (−1/2)^k, 1/(k(k+1)), sin(k), sqrt(k). Powers use ^, products can be written 2k or 2*k, and the functions are sin, cos, tan, ln, log, sqrt, exp, abs with the constants pi and e. Factorials and binomial coefficients are not supported. Both limits are whole numbers (negative allowed, up to 10¹² in size); type inf for an infinite upper limit. Closed forms are used whenever the terms are polynomial or geometric; otherwise up to 2,000 terms are added exactly as fractions (1/(k(k+1)) from 1 to 10 gives 10/11) and up to a million terms as decimals.
Common mistakes and limits
Counting the terms wrongly: from 3 to 7 there are 5 terms, not 4. Using Σ1 = 1 instead of n. Applying the formulas for sums from 1 to n to a sum that starts elsewhere without subtracting the part before the start. Using the geometric formula with the wrong first term when the exponent is k − 1 or k + 2. Concluding that a series converges because its terms go to 0: the harmonic series 1/k has terms that go to 0 and still diverges (OpenStax Calculus Volume 2, section 5.3). The calculator decides convergence only for polynomial, geometric and p-series terms; for any other infinite series it says so and shows just the first partial sums, which do not prove anything.
Sources: OpenStax, Calculus Volume 1, section 5.1 “Approximating Areas” (Examples 5.1, 5.2); OpenStax, Algebra and Trigonometry 2e, section 13.4 “Series and Their Notations” (Examples 1, 2, 4, 7); OpenStax, Calculus Volume 2, sections 5.2 “Infinite Series” (Example 5.9) and 5.3 “The Divergence and Integral Tests”. Related: sequence calculator, Taylor series calculator, integral calculator, scientific calculator and average calculator.