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Summation Calculator – Sum of a Series

Use this free summation calculator to find the sum of an arithmetic or geometric series. Calculate finite series step by step, or find the sum of an infinite geometric series when it converges. The calculator shows the formula, substitution, result, and a preview of the series.

Calculate the Sum of a Series

Choose arithmetic when the terms change by a constant difference. Choose geometric when the terms are multiplied by a constant ratio.
Infinite mode is available for geometric series and requires |r| < 1 for convergence.
The first term of the sequence.
For an arithmetic series, enter the common difference.
Enter a whole number from 1 to 10,000. Infinite series mode does not require a number of terms.

What Is a Summation?

A summation is the process of adding a sequence of numbers together. In mathematics, summation is commonly represented using the Greek capital letter sigma, Σ. A series is the result obtained when the terms of a sequence are added together.

For example, the series 2 + 4 + 6 + 8 + 10 has five terms and its sum is 30.

The formula you need depends on the type of series. Two of the most common types are arithmetic series and geometric series.

Arithmetic Series

An arithmetic series is obtained by adding the terms of an arithmetic sequence. Each consecutive term differs from the previous term by the same constant amount.

If the first term is a, the common difference is d, and there are n terms, the sum is:

Sn = n/2 [2a + (n − 1)d]

An equivalent formula is:

Sn = n(a + l)/2

Here l is the last term of the arithmetic sequence.

Arithmetic series example

Consider:

3 + 7 + 11 + 15 + 19

The first term is a = 3, the common difference is d = 4, and there are n = 5 terms.

S5 = 5/2 [2(3) + (5 − 1)(4)] = 55

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. Instead of adding a constant difference, each term is obtained by multiplying the previous term by the same common ratio.

For a finite geometric series with first term a, common ratio r, and n terms, the formula is:

Sn = a(1 − rn) / (1 − r)

This formula applies when r ≠ 1. When r = 1, every term is equal to the first term, so:

Sn = an

Geometric series example

Consider:

2 + 6 + 18 + 54

Here a = 2, r = 3, and n = 4.

S4 = 2(1 − 34) / (1 − 3) = 80

Infinite Geometric Series

An infinite geometric series has infinitely many terms. Unlike a finite series, its sum exists only when the series converges.

For a geometric series a + ar + ar² + ar³ + …, the convergence condition is:

|r| < 1

When |r| < 1, the infinite sum is:

S∞ = a / (1 − r)

Infinite geometric series example

Consider:

8 + 4 + 2 + 1 + 0.5 + …

The first term is 8 and the common ratio is 1/2. Since |1/2| < 1, the series converges.

S∞ = 8 / (1 − 1/2) = 16

If |r| ≥ 1, an infinite geometric series generally does not have a finite sum. The calculator therefore reports that the series diverges rather than returning a misleading numerical answer.

Arithmetic vs. Geometric Series

Feature Arithmetic Geometric
How terms change Add a constant difference Multiply by a constant ratio
Key value Common difference d Common ratio r
Example 3, 7, 11, 15 3, 6, 12, 24
Finite sum Sn = n/2[2a + (n−1)d] Sn = a(1−rn)/(1−r)
Infinite sum Not generally finite Converges when |r| < 1

Finite Series vs. Infinite Series

Finite series

A finite series contains a specific number of terms. For example:

2 + 4 + 6 + 8

The number of terms is known, so a finite-series formula can be used directly.

Infinite geometric series

An infinite geometric series continues forever:

a + ar + ar² + ar³ + …

A finite sum exists only when the common ratio satisfies |r| < 1, apart from the trivial all-zero series.

How to Use the Summation Calculator

  1. Select Arithmetic series or Geometric series.
  2. Select Finite series or, for a geometric series, Infinite geometric series.
  3. Enter the first term a.
  4. Enter the common difference d for an arithmetic series, or the common ratio r for a geometric series.
  5. For a finite series, enter the number of terms n.
  6. Click Calculate Sum.

The calculator then displays the result together with the relevant mathematical formula and calculation steps.

Frequently Asked Questions

What is the formula for the sum of an arithmetic series?

The standard formula is Sn = n/2[2a + (n − 1)d] , where a is the first term, d is the common difference, and n is the number of terms.

What is the formula for the sum of a geometric series?

For a finite geometric series with r ≠ 1, use Sn = a(1 − rn)/(1 − r) . When r = 1, the sum is simply an.

When does an infinite geometric series converge?

An infinite geometric series converges when |r| < 1. Its sum is a/(1 − r).

Can the common difference be negative?

Yes. Negative common differences are valid. For example, 20, 15, 10, 5 is an arithmetic sequence with d = −5.

Can the common ratio be negative?

Yes. A geometric sequence can have a negative ratio. For example, 4, −2, 1, −0.5 has a common ratio of −1/2.

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers. A series is formed by adding the terms of a sequence. For example, 2, 4, 6, 8 is a sequence, while 2 + 4 + 6 + 8 is a series.

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