Find The Taylor Series Calculator

Calculate Taylor Series Approximation

Enter values and click Calculate.

Series Expression: –

Original f(x) at x: –

Approximation at x: –

Absolute Error: –

The Taylor series approximates f(x) around 'a' as: f(a) + f'(a)(x-a) + f"(a)/2! (x-a)^2 + … + f^(n-1)(a)/(n-1)! (x-a)^(n-1)

What is a Taylor Series Calculator?

A Taylor Series Calculator is a tool used to find the Taylor series expansion of a function around a given point. The Taylor series is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. A finite number of terms from this series can be used to approximate the function near that point. This calculator helps you visualize and compute this approximation for common functions.

It's used by students, engineers, and scientists to approximate functions that might be difficult to compute directly, or to understand the local behavior of a function around a specific point. For instance, many complex functions in physics and engineering are simplified using their Taylor series approximations, especially when considering small changes around an equilibrium or operating point. The Taylor Series Calculator makes these calculations accessible.

Common misconceptions include thinking the Taylor series is always a perfect representation of the function everywhere (it's often best near the expansion point) or that more terms always dramatically improve the approximation far from 'a' (the radius of convergence matters).

Taylor Series Formula and Mathematical Explanation

The Taylor series of a real or complex-valued function f(x) that is infinitely differentiable at a real or complex number 'a' is the power series:

f(x) = ∑_n=0^∞ [f⁽ⁿ⁾(a) / n!] * (x-a)ⁿ

Where:

• f⁽ⁿ⁾(a) is the n-th derivative of f evaluated at the point a.
• n! is the factorial of n.
• (x-a)ⁿ is the difference between x and a raised to the power of n.

The Taylor Series Calculator computes a finite number of terms of this series, known as the Taylor polynomial of degree N-1 if N terms are used (from n=0 to N-1):

P_N-1(x) = f(a) + f'(a)(x-a) + [f"(a)/2!](x-a)² + … + [f^(N-1)(a)/(N-1)!](x-a)^N-1

This polynomial P_N-1(x) approximates f(x) for values of x near 'a'. The more terms included, the better the approximation generally is within the radius of convergence.

Variables Table

Variable	Meaning	Unit	Typical Range
f(x)	The function to be approximated	Varies	e.g., sin(x), exp(x)
a	The point of expansion	Same as x	Real numbers (a>0 for ln(x))
n or k	The order of the derivative/term index	Integer	0, 1, 2, …
x	The point at which to evaluate the approximation	Same as a	Real numbers
N	Number of terms in the polynomial	Integer	1, 2, 3, … (e.g., up to 20 in the calculator)

Practical Examples (Real-World Use Cases)

Example 1: Approximating sin(x) near x=0

Let's approximate f(x) = sin(x) around a=0 using 3 terms (n=0, 1, 2).

• f(x) = sin(x), f(0) = sin(0) = 0
• f'(x) = cos(x), f'(0) = cos(0) = 1
• f"(x) = -sin(x), f"(0) = -sin(0) = 0
• f"'(x) = -cos(x), f"'(0) = -cos(0) = -1

Using 3 terms (up to n=2): sin(x) ≈ 0 + 1*(x-0)/1! + 0*(x-0)^2/2! = x. Using 4 terms (up to n=3): sin(x) ≈ x – x^3/3! = x – x^3/6.

If we use the Taylor Series Calculator with f(x)=sin(x), a=0, 4 terms, and evaluate at x=0.1:

Approximation = 0.1 – (0.1)^3/6 = 0.1 – 0.001/6 ≈ 0.1 – 0.00016667 = 0.09983333

Actual sin(0.1) ≈ 0.09983341

The approximation is very close.

Example 2: Approximating exp(x) near x=0

Let's approximate f(x) = exp(x) around a=0 using 4 terms (n=0, 1, 2, 3).

• f(x) = exp(x), f(0) = exp(0) = 1
• f'(x) = exp(x), f'(0) = exp(0) = 1
• f"(x) = exp(x), f"(0) = exp(0) = 1
• f"'(x) = exp(x), f"'(0) = exp(0) = 1

exp(x) ≈ 1 + 1*(x-0)/1! + 1*(x-0)^2/2! + 1*(x-0)^3/3! = 1 + x + x^2/2 + x^3/6

Using the Taylor Series Calculator with f(x)=exp(x), a=0, 4 terms, and evaluate at x=0.5:

Approximation = 1 + 0.5 + (0.5)^2/2 + (0.5)^3/6 = 1 + 0.5 + 0.25/2 + 0.125/6 = 1 + 0.5 + 0.125 + 0.0208333… ≈ 1.6458333

Actual exp(0.5) ≈ 1.648721

The approximation is quite good.

How to Use This Taylor Series Calculator

1. Select Function: Choose the function f(x) you want to approximate from the dropdown menu (e.g., sin(x), cos(x), exp(x), ln(x)).
2. Enter Expansion Point (a): Input the point 'a' around which the Taylor series is centered. For ln(x), 'a' must be greater than 0.
3. Enter Number of Terms (n): Specify how many terms of the Taylor series you want to use for the approximation (from 1 to 20).
4. Enter Evaluation Point (x): Input the value of 'x' where you want to compare the function and its approximation.
5. Calculate: The calculator automatically updates as you change values, or you can click "Calculate".
6. View Results: The primary result shows the approximated value at 'x', along with the original function's value and the error. The series expression is also displayed.
7. Examine Table and Chart: The table details each term's contribution, and the chart visually compares the function and its Taylor polynomial.
8. Reset: Use the "Reset" button to return to default values.
9. Copy Results: Use "Copy Results" to copy the main output values.

The Taylor Series Calculator helps you understand how the number of terms and the distance (x-a) affect the accuracy of the approximation.

Key Factors That Affect Taylor Series Results

• Number of Terms: Generally, more terms lead to a better approximation near 'a', within the radius of convergence. However, adding terms might not improve accuracy far from 'a'. Our Taylor Series Calculator allows up to 20 terms.
• Point of Expansion (a): The choice of 'a' is crucial. The approximation is usually best near 'a'.
• Evaluation Point (x): The further 'x' is from 'a', the less accurate the approximation might be for a fixed number of terms. The difference (x-a) is raised to increasing powers.
• The Function Itself: Some functions converge rapidly with few terms, while others require many terms for good accuracy, especially if they have rapid changes or singularities.
• Radius of Convergence: For some functions, the Taylor series only converges (and thus accurately represents the function) within a certain range of x values around 'a'. Outside this range, the series may diverge.
• Computational Precision: When many terms are used, or (x-a) is large, floating-point precision limitations can affect the accuracy of the calculated sum. The Taylor Series Calculator uses standard browser floating-point arithmetic.

Frequently Asked Questions (FAQ)

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

A Maclaurin series is a special case of the Taylor series where the point of expansion 'a' is 0. Our Taylor Series Calculator can compute Maclaurin series by setting 'a' to 0.

Why does the approximation get worse when x is far from a?

The Taylor series is a local approximation. Terms involve (x-a)^n. When |x-a| is large, these terms can grow very quickly, and more terms are needed for convergence, or the series might even diverge.

How many terms do I need for a good approximation?

It depends on the function, the distance |x-a|, and the desired accuracy. The Taylor Series Calculator's table and chart can help you see how the approximation improves with more terms.

Can I use this calculator for any function?

This calculator is pre-configured for sin(x), cos(x), exp(x), and ln(x). Approximating arbitrary functions requires symbolic differentiation, which is complex and beyond the scope of this client-side tool.

What if I enter a large number of terms?

The calculator is limited to 20 terms to manage computation time and potential precision issues with very large or small numbers involved in high-order terms.

Why does ln(x) require a > 0?

The natural logarithm ln(x) is only defined for x > 0. Also, its derivatives involve 1/x^n, so we need a > 0 for the derivatives at 'a' to be defined.

What does the chart show?

The chart plots the original function (blue line) and the Taylor polynomial approximation (red line) over a range of x-values centered around 'a', allowing you to visually compare them.

How is the error calculated?

The absolute error is calculated as |Original f(x) – Approximated value| at the given evaluation point x.

Related Tools and Internal Resources

• Derivative Calculator: Useful for finding the derivatives needed for the Taylor series terms.
• Integral Calculator: Explore the inverse operation of differentiation.
• Limit Calculator: Understand the behavior of functions as they approach a point.
• Series Convergence Calculator: Determine if an infinite series converges or diverges.
• Polynomial Calculator: Work with polynomial functions, which are what Taylor series produce as approximations.
• Function Grapher: Plot various functions to visualize their behavior.