Use this calculator to find the Maclaurin series expansion for a given function at a specified point.

Maclaurin Series Calculation Formula

The following formula is used to calculate the Maclaurin series of a function at a given point.

f(x) = f(0) + f'(0)x + f''(0)x^2/2! + ... + f^(n)(0)x^n/n!

Variables:

  • f(x) is the function value at x
  • f(0) is the function value at 0
  • f'(0) is the first derivative of the function at 0
  • f”(0) is the second derivative of the function at 0
  • x is the point of expansion
  • n is the order of the series

To calculate the Maclaurin series, compute the derivatives of the function at 0, then use the formula to find the series expansion.

What is a Maclaurin Series?

The Maclaurin series is a special case of the Taylor series, where the function is expanded at the point a = 0. This series provides an approximation of the function near the point of expansion, using the function’s derivatives at that point. The Maclaurin series is widely used in mathematics , physics, and engineering for simplifying complex functions and solving differential equations.

How to Calculate the Maclaurin Series?

The following steps outline how to calculate the Maclaurin series for a given function.

  1. Determine the function and the point of expansion (a = 0).
  2. Compute the function’s derivatives at the point of expansion.
  3. Use the formula: f(x) = f(0) + f'(0)x + f”(0)x^2/2! + … + f^(n)(0)x^n/n!
  4. Insert the computed derivatives into the formula to find the series expansion.
  5. Verify your result using the Maclaurin series calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Function: f(x) = e^x

Point of Expansion: a = 0

Order of Series: n = 5

FAQ

1. What is a Maclaurin series?

The Maclaurin series is a type of Taylor series expansion of a function about 0. It is used to approximate functions using the sum of their derivatives at 0.

2. How is the Maclaurin series different from the Taylor series?

The Maclaurin series is a special case of the Taylor series where the expansion point is a = 0. In contrast, the Taylor series can expand a function around any point a.

3. How often should I use the Maclaurin series calculator?

It’s helpful to use the Maclaurin series calculator whenever you need to approximate a function near 0, especially when dealing with complex functions or differential equations.

4. Can this calculator handle higher-order derivatives?

Yes, the calculator can handle higher-order derivatives. You can input the derivatives up to the desired order to get the series expansion.

5. Is the calculator accurate?

The calculator provides an accurate approximation of the Maclaurin series based on the inputs provided. For exact figures, it’s best to manually verify the derivatives and computations.