Series Calculator – Easy Arithmetic, Geometric, Power & Taylor Series

Calculate the sum and terms of arithmetic, geometric, power, and Taylor series with our free Series Calculator. Enter the required values to find series sums, individual terms, Taylor polynomial approximations, and step-by-step calculations.

Whether you need to calculate a finite arithmetic or geometric series, evaluate a power series, or approximate a function with a Taylor polynomial, this calculator provides the result along with the calculation steps.

What Is a Series?

A series is the sum of the terms of a sequence. For example:

2 + 5 + 8 + 11 + 14 = 40

The terms follow a specific pattern, and a series calculator can help determine their sum without calculating every step manually.

Depending on the pattern, a series may be arithmetic, geometric, a power series, or a Taylor series.

How to Use the Series Calculator:

Use the appropriate calculator tab for the type of series you want to calculate:

1. Arithmetic Series — enter the first term, common difference, and number of terms.
2. Geometric Series — enter the first term, common ratio, and number of terms.
3. Power Series — enter the coefficients, center, and value of x.
4. Taylor Series — select the function, center, x-value, and polynomial degree.
5. Click Calculate Series to see the result.
6. Review the series terms and calculation steps shown in the results.

Arithmetic Series Calculator:

An arithmetic series is formed by adding the terms of an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms remains constant.

For example:

2 + 5 + 8 + 11 + 14

Here:
• First term: a₁ = 2
• Common difference: d = 3
• Number of terms: n = 5

The sum is:

For this example:

$S₅ = 5/2 [2(2) + (5 − 1)(3)] S₅ = 40$

Our calculator shows the terms, final sum, last term, and calculation steps.

Geometric Series Calculator:

A geometric series is a series in which each term is obtained by multiplying the previous term by the same common ratio.

For example:

12 + 6 + 3 + 1.5 + 0.75

Here:
• First term: a₁ = 12
• Common ratio: r = 0.5
• Number of terms: n = 5

For a finite geometric series where r ≠ 1, the sum is:

$Sₙ = a₁(1 − rⁿ)/(1 − r)$

For the example above:

$S₅ = 12(1 − 0.5⁵)/(1 − 0.5) S₅ = 23.25$

If the common ratio is r = 1, every term is the same and the finite sum is simply:

$Sₙ = n × a₁$

Infinite Series Calculator:

An infinite series continues indefinitely, so its sum depends on whether the series converges or diverges.

For an infinite geometric series:

$a₁ + a₁r + a₁r² + a₁r³ + …$

the series converges when:

$|r| < 1$

Its infinite sum is:

$S∞ = a₁/(1 − r)$

If |r| ≥ 1, the infinite geometric series does not have a finite sum and is considered divergent.

Our calculator handles finite geometric sums and identifies the convergence condition for an infinite geometric series.

Power Series Calculator:

A power series represents a function as a sum of powers of a variable.

A general power series centered at c has the form:

$Σ aₙ(x − c)ⁿ$

For example:

$a₀ + a₁(x − c) + a₂(x − c)² + a₃(x − c)³ + …$

Our Power Series Calculator lets you enter the coefficients, center, and value of x to evaluate the resulting polynomial.

For example, if the coefficients are:

1, 2, 3

and the center is:

c = 0

the expression becomes:

1 + 2x + 3x²

The calculator evaluates the expression for the x-value you provide.

Important: A finite list of coefficients is enough to evaluate the polynomial, but it is not generally enough to determine the radius of convergence of an infinite power series.

Taylor Series Calculator:

A Taylor series represents a function as an infinite sum of derivatives evaluated around a center point.

The Taylor expansion of a function f(x) around x = a is:

$f(x) = f(a) + f′(a)(x − a) + f″(a)/2!(x − a)² + …$

A Taylor polynomial uses only a finite number of terms to approximate the function.

Our Taylor Series Calculator allows you to choose a function, enter the center a, select an x-value, and specify the polynomial degree.

For example, the Maclaurin series is a Taylor series centered at:

a = 0

The calculator provides the Taylor polynomial approximation and compares the approximation with the function’s reference value.

Taylor Series vs. Maclaurin Series:

A Taylor series can be centered at any value a.

A Maclaurin series is simply a Taylor series centered at zero:

a = 0

For example, the Maclaurin expansion of sine begins:

$sin(x) = x − x³/3! + x⁵/5! − …$

Changing the center allows a Taylor polynomial to approximate a function around a different point.

Series Convergence:

Convergence describes whether the terms of an infinite series approach a finite sum.

A series can be:
• Convergent — approaches a finite value.
• Divergent — does not approach a finite value.
• Conditionally convergent — converges under a particular interpretation or condition.
• Absolutely convergent — the series remains convergent when absolute values of its terms are used.

Different types of series require different convergence tests. Common tests include the ratio test, comparison test, root test, and integral test.

For a geometric series, the convergence rule is particularly simple:

|r| < 1 → convergent
|r| ≥ 1 → divergent

Common Series Formulas:

Arithmetic Series:

$Sₙ = n/2 [2a₁ + (n − 1)d]$

Where:
• a₁ = first term
• d = common difference
• n = number of terms

Geometric Series:

Sₙ = a₁(1 − rⁿ)/(1 − r)

Where:
• a₁ = first term
• r = common ratio
• n = number of terms

Infinite Geometric Series:

S∞ = a₁/(1 − r)

This formula applies when |r| < 1.

Power Series

Σ aₙ(x − c)ⁿ

Taylor Series:

Σ [f⁽ⁿ⁾(a)/n!] (x − a)ⁿ

Examples of Series Calculations:

Example 1: Arithmetic Series

Calculate:

4 + 7 + 10 + 13 + 16

The sum is:

50

Example 2: Geometric Series

Calculate:

3 + 6 + 12 + 24

The sum is:

45

Example 3: Infinite Geometric Series

Calculate:

10 + 5 + 2.5 + 1.25 + …

Because r = 0.5 and |r| < 1, the series converges.

Its sum is:

10/(1 − 0.5) = 20

Example 4: Power Series

For:

1 + 2x + 3x²

at:

x = 2

the value is:

1 + 4 + 12 = 17

Infinite series are an important part of calculus, where derivatives and limits are frequently used. Use our Derivative Calculator to calculate derivatives and understand calculus concepts.

FAQ

What is a series calculator?

A series calculator is a tool used to calculate the terms and sums of different types of mathematical series, including arithmetic and geometric series. It can also evaluate power series and Taylor polynomial approximations.

How do you calculate the sum of a series?

The method depends on the type of series. Arithmetic and geometric series have specific sum formulas, while other series may require convergence tests, partial sums, or other mathematical techniques.

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers. A series is the sum of the terms in a sequence.

How do you find the sum of an infinite series?

First determine whether the series converges. For an infinite geometric series with |r| < 1, use S∞ = a₁/(1 − r). Other infinite series require appropriate convergence tests and formulas.

What is a power series?

A power series is an infinite expression involving powers of a variable, commonly written as Σ aₙ(x − c)ⁿ.

What is a Taylor series?

A Taylor series represents a function using its derivatives around a chosen center point. A finite number of terms produces a Taylor polynomial approximation.

Is a Maclaurin series a Taylor series?

Yes. A Maclaurin series is a Taylor series whose center is zero.

To learn more about arithmetic series, geometric series, power series, and advanced mathematical concepts, visit Wolfram MathWorld – Series.