Sum of Series Calculator
Enter your series parameters to find the exact sum, whether your series is arithmetic, geometric, or one of several other common types. The calculator shows the closed-form formula, substitutes your values into every step, lists all the individual terms, and draws a cumulative-sum chart so you can see how the total builds up or converges.
What is a series?
A series is the sum of the terms of a sequence. If a sequence lists the values a1, a2, a3, ..., the corresponding series adds them up: S = a1 + a2 + a3 + ... A finite series stops after a fixed number of terms n, while an infinite series continues forever - sometimes reaching a finite limit called the sum (convergent), and sometimes growing without bound (divergent). The six types in this calculator cover the most common cases: arithmetic progressions, geometric progressions, sums of squares, sums of cubes, and the harmonic series.
Arithmetic series
An arithmetic series is formed by adding a constant difference d to each successive term. If the first term is a1, the terms are a1, a1+d, a1+2d, and so on. The closed-form formula for the partial sum of n terms is Sn = n/2 x (2a1 + (n-1)d), which is equivalent to n times the average of the first and last terms. For example, 1 + 3 + 5 + 7 + 9 (five terms starting at 1 with d = 2) sums to 5/2 x (2 + 8) = 25. An arithmetic series diverges as n goes to infinity unless both a1 and d are zero.
Geometric series - finite and infinite
In a geometric series each term is multiplied by a fixed ratio r. For n finite terms starting at a1 the sum is Sn = a1(1 - r^n)/(1 - r) when r is not equal to 1. When |r| is strictly less than 1 each successive term shrinks, and as n goes to infinity the partial sums converge to the infinite sum S = a1/(1 - r). For example, 1 + 0.5 + 0.25 + ... converges to 1/(1 - 0.5) = 2. When |r| is greater than or equal to 1 the terms do not shrink and the series diverges.
Sum of squares, cubes, and harmonic series
The sum of the first n perfect squares, 1 + 4 + 9 + ... + n^2, has the elegant closed form n(n+1)(2n+1)/6. The sum of the first n perfect cubes, 1 + 8 + 27 + ... + n^3, equals [n(n+1)/2]^2 - the square of the n-th triangular number, a beautiful identity discovered in antiquity. The harmonic series 1 + 1/2 + 1/3 + ... + 1/n grows without bound, but extremely slowly: it exceeds 10 only around n = 12,000 and exceeds 20 only around n = 272 million. All three are divergent as n goes to infinity, but each has a useful partial sum for finite n.
Common series formulas
| Series type | n-th term | Sum formula | Converges? |
|---|---|---|---|
| Arithmetic | a + (k-1)d | n/2 x (2a + (n-1)d) | Only if a=d=0 |
| Geometric (finite) | a x r^(k-1) | a(1-r^n)/(1-r) | Partial sum |
| Geometric (infinite) | a x r^(k-1) | a/(1-r) | Yes, when |r| < 1 |
| Sum of squares | k^2 | n(n+1)(2n+1)/6 | No (diverges) |
| Sum of cubes | k^3 | [n(n+1)/2]^2 | No (diverges) |
| Harmonic | 1/k | ln(n) + gamma | No (diverges) |
Closed-form results for the most frequently used finite and infinite series.
Frequently asked questions
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers, such as 1, 2, 4, 8, 16. A series is the sum of a sequence: 1 + 2 + 4 + 8 + 16 = 31. This calculator computes series, meaning it adds up the terms rather than just listing them.
When does a geometric series converge?
An infinite geometric series converges - meaning it has a finite sum - when the absolute value of the common ratio |r| is strictly less than 1. If |r| is greater than or equal to 1 the terms do not shrink fast enough and the series diverges. For example, 1 + 0.5 + 0.25 + ... (r = 0.5) converges to 2, but 1 + 2 + 4 + ... (r = 2) grows without bound.
What formula gives the sum of an arithmetic series?
The standard formula is Sn = n/2 x (2a1 + (n - 1)d), where a1 is the first term, d is the common difference, and n is the number of terms. An equivalent form is Sn = n x (a1 + an)/2, the number of terms times the average of the first and last term. Both give the same answer and each is useful in different contexts.
Does the harmonic series 1 + 1/2 + 1/3 + ... converge?
No. Despite each additional term getting smaller, the harmonic series diverges - it has no finite sum. The proof, dating back to Nicole Oresme around 1350, groups terms to show the sum exceeds arbitrarily large integers. However, partial sums for finite n are well-defined and useful in probability, music theory, and computer science.
Why does the sum of cubes equal the square of the sum of integers?
The identity 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2 = [n(n+1)/2]^2 can be proved by mathematical induction. It means the sum of the first n cubes is always a perfect square. This is sometimes called Nicomachus's theorem after the ancient Greek mathematician who stated it around 100 CE.
How accurate is the calculator for large n?
For arithmetic, squares, and cubes the closed-form formula is exact regardless of n. For geometric series the formula is exact in exact arithmetic, but floating-point rounding accumulates for very large n or ratios very close to 1. For harmonic series the calculator iterates term by term, which accumulates small floating-point errors for very large n. For everyday use up to several hundred terms the results are accurate to many decimal places.