What is a Taylor series?

A Taylor series is an infinite sum of terms representing a function as a polynomial, calculated using the function's derivatives at a single point (the center).

What is the difference between a Taylor and a Maclaurin series?

A Taylor series can be centered at any point $a$. A Maclaurin series is simply a Taylor series centered specifically at $a = 0$.

Why are Taylor series useful?

They allow complex transcendental functions (like trigonometric and exponential functions) to be approximated by polynomials, which are much easier to calculate, integrate, and differentiate.

What is a Taylor polynomial?

A Taylor polynomial is a truncated (finite) version of the infinite Taylor series, representing the approximation up to a specific degree $n$.

What does convergence mean for a Taylor series?

Convergence means that as the number of terms increases, the sum of the series approaches the exact value of the function. The set of values for which this happens is the interval of convergence.

How do you calculate the remainder or error of a Taylor approximation?

The remainder is calculated using Taylor's theorem (often the Lagrange form), which estimates the maximum difference between the function and the Taylor polynomial.

Can any function be represented by a Taylor series?

No, only functions that are infinitely differentiable in a neighborhood around the center point can have a Taylor series. Such functions are called analytic.

What happens if you evaluate a Taylor series outside its interval of convergence?

The terms of the series will grow infinitely (diverge) and the sum will not equal the function's actual value.

How many terms are needed for a good Taylor series approximation?

The number of terms depends on the function, the distance from the center point, and the desired accuracy. Functions evaluated close to the center require fewer terms.

What is the Maclaurin series for e^x?

The Maclaurin series for $e^x$ is $1 + x + x^2/2! + x^3/3! + x^4/4! + \dots$, which converges for all real numbers $x$.

What is the Maclaurin series for sin(x)?

The Maclaurin series for $\sin(x)$ consists of only odd powers: $x - x^3/3! + x^5/5! - x^7/7! + \dots$, which converges for all real numbers $x$.

What is the Maclaurin series for cos(x)?

The Maclaurin series for $\cos(x)$ consists of only even powers: $1 - x^2/2! + x^4/4! - x^6/6! + \dots$, which converges for all real numbers $x.

math calculator

Taylor & Maclaurin Series Calculator

Compute Taylor and Maclaurin polynomial series approximations with step-by-step error analysis.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Function f(x)

Center Point a

Polynomial Degree n (1 – 8)

Evaluation Point x

Results

P4(1) Approximation

2.708333

Exact Value2.718282

Absolute Error9.9485e-3

Relative Error0.3660%

Convergence Curve

Blue: exact f(x) | Dashed: Taylor approximation Pₙ(x)

Step-by-Step Expansion Steps

For $f(x) = e^x$, all derivatives are $f^{(k)}(x) = e^x$. At center $a = 0$, $f^{(k)}(0) = e^{0} \approx 1.0000$.

The Taylor expansion is constructed as:

\[P_{4}(1) = 1.0000 + \frac{1.0000}{1!}(1.00)^{1} + \frac{1.0000}{2!}(1.00)^{2} + \frac{1.0000}{3!}(1.00)^{3} + \frac{1.0000}{4!}(1.00)^{4}\]

\[P_{4}(1) \approx 2.708333\]

Formula

Taylor & Maclaurin Series formulas

P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^k, \quad R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}

A Taylor polynomial centers the polynomial expansion around point a. The remainder term R_n(x) defines the Lagrange error bound.

Step by step

How the calculation works

1Select the base function type (Exponential, Sine, Cosine, or Natural Logarithm).
2Specify the center point (a) for expansion and the target approximation degree (n).
3Enter the point (x) to evaluate. The calculator expands terms, evaluates the summation, and calculates the absolute error.

Related guides

Learn more about this calculation

Guide

Scientific Notation Guide: Powers of Ten Made Practical

Learn how to convert, compare, and calculate with powers of ten using worked examples and the live DTC scientific calculator.

Site Guide

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Learn the fastest reliable workflow for using Do The Calculation calculators, reading results, checking formulas, and using save, print, share, and export actions correctly.

Scientific

Scientific Calculator

Evaluate transcendental functions directly.

Gamma Func

Factorial & Gamma Calculator

Verify factorial terms used in Taylor denominators.

math calculator

Taylor & Maclaurin Series Calculator

Compute Taylor and Maclaurin polynomial series approximations with step-by-step error analysis.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Function f(x)

Center Point a

Polynomial Degree n (1 – 8)

Evaluation Point x

Results

P4(1) Approximation

2.708333

Exact Value2.718282

Absolute Error9.9485e-3

Relative Error0.3660%

Convergence Curve

Blue: exact f(x) | Dashed: Taylor approximation Pₙ(x)

Step-by-Step Expansion Steps

For $f(x) = e^x$, all derivatives are $f^{(k)}(x) = e^x$. At center $a = 0$, $f^{(k)}(0) = e^{0} \approx 1.0000$.

The Taylor expansion is constructed as:

\[P_{4}(1) = 1.0000 + \frac{1.0000}{1!}(1.00)^{1} + \frac{1.0000}{2!}(1.00)^{2} + \frac{1.0000}{3!}(1.00)^{3} + \frac{1.0000}{4!}(1.00)^{4}\]

\[P_{4}(1) \approx 2.708333\]

Formula

Taylor & Maclaurin Series formulas

P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^k, \quad R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}

A Taylor polynomial centers the polynomial expansion around point a. The remainder term R_n(x) defines the Lagrange error bound.

Step by step

How the calculation works

1Select the base function type (Exponential, Sine, Cosine, or Natural Logarithm).
2Specify the center point (a) for expansion and the target approximation degree (n).
3Enter the point (x) to evaluate. The calculator expands terms, evaluates the summation, and calculates the absolute error.

Related guides

Learn more about this calculation

Guide