Use a known Maclaurin series to obtain the Maclaurin series for the function f(x)
= cos(πx).

To find the Maclaurin series for the function \( f(x) = \cos(\pi x) \), we can use the known Maclaurin series for \( \cos(x) \).

### Step 1: Recall the Maclaurin series for \( \cos(x) \)

The Maclaurin series for \( \cos(x) \) is given by:

\[
\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}
\]

This series converges for all \( x \).

### Step 2: Substitute \( \pi x \) for \( x \)

To find the Maclaurin series for \( \cos(\pi x) \), we substitute \( x \) with \( \pi x \) in the series:

\[
\cos(\pi x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} (\pi x)^{2n}
\]

### Step 3: Simplify the expression

We can now simplify the expression by separating the powers of \( \pi \) and \( x \):

\[
\cos(\pi x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \pi^{2n} x^{2n}
\]

### Step 4: Write the Maclaurin series explicitly

So, the Maclaurin series for \( f(x) = \cos(\pi x) \) is:

\[
\cos(\pi x) = 1 - \frac{(\pi x)^2}{2!} + \frac{(\pi x)^4}{4!} - \frac{(\pi x)^6}{6!} + \cdots
\]

Or, more explicitly:

\[
\cos(\pi x) = 1 - \frac{\pi^2 x^2}{2} + \frac{\pi^4 x^4}{24} - \frac{\pi^6 x^6}{720} + \cdots
\]

This is the Maclaurin series for \( \cos(\pi x) \).

### Summary

The Maclaurin series for \( \cos(\pi x) \) is:

\[
\cos(\pi x) = \sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n} x^{2n}}{(2n)!}
\]

Would you like more details or have any questions?

### Related Questions:

1. What is the Maclaurin series for \( \sin(x) \) and how can it be used to find the series for \( \sin(\pi x) \)?
2. How do we determine the radius of convergence for a Maclaurin series?
3. Can the Maclaurin series for \( \cos(x) \) be used to approximate \( \cos(\pi/4) \) to a certain number of decimal places?
4. How can we use the Maclaurin series for \( \cos(x) \) to derive the series for \( e^x \)?
5. What is the general formula for finding the Maclaurin series of a given function?

### Tip:

When working with Maclaurin series, substituting variables can help to quickly find the series for transformed functions.

Use a known Maclaurin series to obtain the Maclaurin series for the function f(x) = cos(πx).

Discover how to derive the Maclaurin series for the function f(x) = cos(πx) by using the known Maclaurin series for cos(x). This detailed step-by-step solution covers key concepts in calculus, including series expansion and the Maclaurin series theorem. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation