What is a geometric series?

A geometric series is a sum of terms where each term is multiplied by a fixed common ratio r to get the next term. The first n terms sum to a times (1 minus r to the power n) divided by (1 minus r), for r not equal to 1. If r is between -1 and 1 the series converges as n grows large; if |r| is 1 or more the series diverges.

What is an arithmetic series?

An arithmetic series is a sum of terms where each term increases by a fixed common difference d. The sum of the first n terms is n times (2a plus (n-1)d) divided by 2, which can also be written as n times the average of the first and last term. Arithmetic series appear in problems involving evenly spaced quantities such as integer sums or regular payment schedules.

When does a geometric series converge to infinity?

A geometric series with first term a and common ratio r has an infinite sum of a divided by (1 minus r) only when the absolute value of r is strictly less than 1. For |r| equal to or greater than 1, adding more terms keeps growing the sum without bound, so there is no finite limit.

Series Sum Calculator

A series is the sum of the terms in a sequence, and two of the most important types appear throughout mathematics, finance, and the physical sciences. An arithmetic series adds up terms that increase by a fixed amount each step, called the common difference, like 1 + 3 + 5 + 7. A geometric series multiplies each term by a fixed ratio to get the next one, like 1 + 0.5 + 0.25 + 0.125. Geometric series appear in compound interest calculations, radioactive decay, signal processing, and probability. Arithmetic series appear in problems involving equally spaced values, such as the total of the first n positive integers or the cost of items priced at equal increments. This calculator handles both types. Select the series type, enter the first term a, the common ratio r (for geometric) or common difference d (for arithmetic), and the number of terms n. The calculator returns the partial sum, the value of the last term, the infinite sum if the geometric series converges, and the formula used. The geometric default uses a = 1, r = 0.5, n = 10 giving a sum of 1.9990. The arithmetic default uses a = 1, d = 2, n = 10 giving a sum of 100. Switch between the two types using the selector below.

Series type

First term (a)

Common ratio (r)

Number of terms (n)

1.9980

partial sum (S_n)

Last term (a_n)0.0020

Infinite sum2.0000

Formulaa(1-r^n)/(1-r)

Infinite sum is shown only when |r| < 1 for geometric series. Arithmetic series have no finite infinite sum.

How it works

For a geometric series with first term a, common ratio r, and n terms: the partial sum is a * (1 - r^n) / (1 - r) when r is not 1, or a * n when r = 1. The last term is a * r^(n-1). The infinite sum a / (1 - r) exists only when |r| is strictly less than 1. For an arithmetic series with first term a, common difference d, and n terms: the partial sum is n * (2a + (n-1) * d) / 2. The last term is a + (n-1) * d. There is no finite infinite sum for an arithmetic series.

Worked example

Geometric series: a = 1, r = 0.5, n = 10. Sum = 1 * (1 - 0.5^10) / (1 - 0.5) = 0.9990234375 / 0.5 = 1.9980. Last term = 1 * 0.5^9 = 0.0020. Infinite sum = 1 / (1 - 0.5) = 2.0000. Arithmetic series (switch type): a = 1, d = 2, n = 10. Sum = 10 * (2 + 9 * 2) / 2 = 10 * 20 / 2 = 100. Last term = 1 + 9 * 2 = 19.

Related calculators

Riemann Sum Calculator: sum rectangular areas to approximate an integral.
Area Under the Curve Calculator: compute integrals using Simpson's rule.
Percent Error Calculator: measure how close a partial sum approximation is to the true value.
Critical Points Calculator: find turning points of a function.

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