How to find the Maclaurin series of a function?
Understand the Problem
The question is asking about the procedure to derive the Maclaurin series representation of a given function. The Maclaurin series is a special case of the Taylor series, centered at zero, which expresses the function as an infinite sum of terms calculated from the values of its derivatives at a single point.
Answer
Maclaurin series
Answer for screen readers
The final answer is the Maclaurin series
Steps to Solve
- Identify the function and its derivatives at zero
The goal is to write the given function as a sum of its derivatives evaluated at zero. Start by computing the function and its derivatives at $x = 0$: $$ f(0), f'(0), f''(0), f'''(0), ext{etc.} $$
- Set up the general formula for the Maclaurin series
A Maclaurin series for a function $f(x)$ is represented as: $$ f(x) = f(0) + f'(0)x + \frac{{f''(0)}}{2!}x^2 + \frac{{f'''(0)}}{3!}x^3 + \cdots $$
- Compute the individual terms of the series
Use the derivatives you computed in step 1 and substitute them into the series formula to get the terms: $$ f(0), f'(0)x, \frac{{f''(0)}}{2!}x^2, \cdots $$
- Write out the Maclaurin series
Combine the terms you calculated: $$ f(x) = f(0) + f'(0)x + \frac{{f''(0)}}{2!}x^2 + \frac{{f'''(0)}}{3!}x^3 + \cdots $$
Make sure to include as many terms as required for the desired approximation accuracy.
The final answer is the Maclaurin series
More Information
The Maclaurin series is useful for approximating functions and solving differential equations.
Tips
A common mistake is not calculating enough terms for the desired accuracy, or incorrectly evaluating the derivatives at zero.
AI-generated content may contain errors. Please verify critical information
