Chapter 8: Q 75. (page 702)

Let $f$ be an odd function with Maclaurin series representation $\sum_{k = 0}^{\infty} a_{k} x^{k}$ . Prove that $a_{2 k} = 0$ for every nonnegative integer $k$ .

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

It is given that function is an odd function.

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

Step 2. Prove the given statement.

It is given that the function $f (x)$ is odd then we have $f (x) = - f (- x)$ .

So, evaluate $f (- x)$ .

role="math" localid="1650443571892" $\begin{array}{rcl} f (- x) & = & \sum_{k = 0}^{\infty} a_{k} {(- x)}^{k} \\ = & \sum_{k = 0}^{\infty} {(- 1)}^{k} a_{k} x^{k} \end{array}$

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

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

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

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Let $f$ be an odd function with Maclaurin series representation $\sum_{k = 0}^{\infty} a_{k} x^{k}$ . Prove that $a_{2 k} = 0$ 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 odd function with Maclaurin series representation ∑k=0∞akxk. Prove that a2k=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

Calculus

Logic and Functions

Pure Maths

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Let $f$ be an odd function with Maclaurin series representation $\sum_{k = 0}^{\infty} a_{k} x^{k}$ . Prove that $a_{2 k} = 0$ for every nonnegative integer $k$ .