Chapter 8: Q 74. (page 702)

Let $f$ be an even function with Maclaurin series representation $\sum_{k = 0}^{\infty} a_{k} x^{k}$ . Prove thatrole="math" localid="1650365708378" $a_{2 k + 1} = 0$ for every nonnegative integer $k$ .

Short Answer

Expert verified

The solution is $a_{2 k + 1} = 0$ for every nonnegative integer $k$ .

Step by step solution

Step 1. Given information.

The given even function is:

$f (x) = \sum_{k = 0}^{\infty} a_{k} x^{k}$

Step 2. Prove the given statement.

It is given that $f (x)$ is an even function then $f (x) = f (- x)$ .

Now evaluate $f (- x)$ .

role="math" localid="1650366151821" $\begin{array}{rcl} f (- x) & = & \sum_{k = 0}^{\infty} a_{k} {(- x)}^{k} \\ = & \sum_{k = 0}^{\infty} (- 1)^{k} a_{k} x^{k} \end{array}$

Since $f (x) = f (- x)$ , implies that $\sum_{k = 0}^{\infty} a_{k} x^{k} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \sum_{k = 0}^{\infty} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} ( \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} 1 \end{array} \begin{array}{rcl} )^{k} \end{array} \begin{array}{rcl} a_{k} \end{array} \begin{array}{rcl} x^{k} \end{array}$ .

That is, $a_{k} = {(- 1)}^{k} a_{k}$

We know that it is possible only when $a_{2 k + 1} = 0$ , for every value of $k$ .

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Let $f$ be an even function with Maclaurin series representation $\sum_{k = 0}^{\infty} a_{k} x^{k}$ . Prove thatrole="math" localid="1650365708378" $a_{2 k + 1} = 0$ for every nonnegative integer $k$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Prove the given statement.

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Let fbe an even function with Maclaurin series representation∑k=0∞akxk. Prove thatrole="math" localid="1650365708378" a2k+1=0for every nonnegative integerk.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Prove the given statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Theoretical and Mathematical Physics

Probability and Statistics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Let $f$ be an even function with Maclaurin series representation $\sum_{k = 0}^{\infty} a_{k} x^{k}$ . Prove thatrole="math" localid="1650365708378" $a_{2 k + 1} = 0$ for every nonnegative integer $k$ .