# What is the sum of a series?

Sums a(n) from n = n0 to a whole number or to infinity, and says whether an infinite series converges.

- Page: https://www.acalculator.org/math/series-calculator
- JSON spec: https://www.acalculator.org/math/series-calculator.json
- Version: b6417e4250d5

## Default answer

Example with the default inputs (Term a(n) (-3)^(n + 1)/4^(n - 1), From n = 1, To n = inf): The sum of (-3)^(n + 1)/4^(n - 1) from n = 1 to inf is 5.142857143.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | Term a(n) | The n-th term, such as 1/(n (n + 1)). |
| from | From n = | The first n. |
| to | To n = | The last n, or inf. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| verdict | The series | Converges or diverges. |
| sum | Sum | The sum of the terms. |
| exact | Exact sum | The sum written exactly. |

## Method

Geometric series with ratio r: a/(1 − r) when |r| < 1. Rational terms: partial fractions that telescope.

## Assumptions

- n takes every whole number from the first to the last.

## Worked examples

1. a = (-3)^(n + 1)/4^(n - 1), from = 1, to = inf gives verdict = Converges, sum = 5.142857, exact = 36/7. Source: OpenStax, Calculus Volume 2, section 5.2 Infinite Series, Example 5.9(a). https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series.
2. a = 1/(n (n + 1)), from = 1, to = inf gives verdict = Converges, sum = 1, exact = 1.
3. a = n/(n + 1), from = 1, to = inf gives verdict = Diverges.

## FAQ

### What is the sum of a geometric series?

A geometric series a + ar + ar² + … has a constant ratio r between terms. If |r| < 1 it converges to a/(1 − r), where a is the first term; if |r| ≥ 1 (and a ≠ 0) it diverges. Σ from n = 1 to ∞ of (1/2)ⁿ = (1/2)/(1 − 1/2) = 1.

### What is a telescoping series?

One whose partial sums collapse because each term cancels part of the next. 1/(n(n + 1)) = 1/n − 1/(n + 1), so the sum from 1 to N is 1 − 1/(N + 1), and the infinite sum is 1.

### How do I know if a series diverges?

If the terms do not go to 0, the series diverges (the divergence test): Σ n/(n + 1) diverges because its terms approach 1. Terms that do go to 0 are not enough: the harmonic series Σ 1/n diverges although 1/n → 0.

### Why does the page say "Converges" but give no sum?

Some series converge to a number with no simple formula the page can find. Σ 1/n² converges (its terms are smaller than those of a telescoping series), and its sum π²/6 is a famous result, but it is not geometric or telescoping, so the page reports convergence only.

### How do I type the term?

Use n as the variable: 1/(n (n + 1)), (1/2)^n, (-1)^n/3^n, n^2. For a finite sum, type the last n; for an infinite series, type inf.

### How is the answer checked?

A finite sum is added term by term, as a decimal. An infinite sum is compared with the partial sums of 10,000 and 100,000 terms, which must approach it.

## Sources

- OpenStax, Calculus Volume 2, section 5.2 Infinite Series: https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series
- OpenStax, Calculus Volume 2, section 5.4 Comparison Tests: https://openstax.org/books/calculus-volume-2/pages/5-4-comparison-tests