Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The Taylor series for an even function centered at 0 has only even powers of x.

Verified step by step guidance

1. Recall the definition of an even function: a function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain.
2. Consider the Taylor series of \( f(x) \) centered at 0, which is given by \( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \).
3. Substitute \( -x \) into the Taylor series: \( f(-x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-x)^n = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-1)^n x^n \).
4. Since \( f \) is even, \( f(-x) = f(x) \), so the series must satisfy \( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-1)^n x^n = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \). This implies that terms with odd powers \( n \) must have zero coefficients because \( (-1)^n = -1 \) for odd \( n \), which would otherwise change the sign.
5. Therefore, the Taylor series for an even function centered at 0 contains only even powers of \( x \), confirming the statement is true.

Key Concepts

Even Functions

An even function satisfies f(-x) = f(x) for all x in its domain. This symmetry about the y-axis means the function's graph is mirrored on both sides, influencing the behavior of its Taylor series expansion.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. Centering at zero (Maclaurin series) expresses the function as powers of x, revealing patterns in coefficients based on the function's properties.

Parity of Terms in Taylor Series for Even Functions

For even functions centered at zero, all odd-powered terms in the Taylor series vanish because the derivatives of odd order at zero are zero. This results in a series containing only even powers of x, reflecting the function's symmetry.