# Thread: Proof of Theorem involving Maclaurin Polynomials

1. ## Proof of Theorem involving Maclaurin Polynomials

I'm trying to understand the proof found here:

Calculus 9th edition, chapter 9, section 7, exercise 71
Calc Chat Free Solutions

The theorem is:
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If $\displaystyle f$ is an even function, then it's nth Maclaurin polynomial contains only terms with even powers of x.

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I don't understand why the coefficients of the terms are zero for the odd derivatives.

2. Which part of the proof do you get hung up on?

3. Originally Posted by Ackbeet
Which part of the proof do you get hung up on?
I don't understand why the coefficients of the terms are zero for the odd derivatives.

4. Take a MacLaurin series: $\displaystyle f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}\,x^{2}+\dots$

If you take the first derivative of this, you're going to make the constant term disappear, and you'll get $\displaystyle f'(0)+f''(0)x+\dots$

If the $\displaystyle f'(0)$ is not zero, then in general, you can't have $\displaystyle f'(x)$ be an odd function (which it must be, since it is the derivative of an even function). Get the idea?