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Chapter 8: Q 75. (page 702)

Let $f$ be an odd function with Maclaurin series representation ${鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$ . 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

01

Step 1. Given information.

It is given that function is an odd function.

${f (x) = 鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$

02

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" $\begin{array}{rcl} f (- x) & = & {鈭�}_{k = 0}^{鈭�} a_{k} {(- x)}^{k} \\ = & {鈭�}_{k = 0}^{鈭�} {(- 1)}^{k} a_{k} x^{k} \end{array}$

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

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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Most popular questions from this chapter

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of $x_{0}$ . Note: These are the same functions and values as in Exercises 41鈥�48.

role="math" localid="1649406173168" $49 . \cos (x), \frac{蟺}{2}$

If $f (x) = 4 x^{3} - 5 x^{2} + 6 x + 1 a n d P_{3} (x)$ is the third Taylor polynomial for f at 鈭�1, what is the third remainder $R_{3} (x)$ ? What is $R_{4} (x)$ ? (Hint: You can answer this question without finding any derivatives.)

Show that the series: $\ln (2) + {鈭�}_{k = 1}^{鈭�} \frac{{(- 1)}^{k - 1}}{k 2^{k}} {(x - 2)}^{k}$

from Example 3 diverges when x = 0 and converges conditionally when x = 4.

How may we find the Maclaurin series for f(x)g(x) if we already know the Maclaurin series for the functions f(x) and g(x)? How do you find the interval of convergence for the new series?

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x :

$J_{p} (x) = {鈭�}_{k = 0}^{鈭�} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

What is the interval of convergence for $J_{1} (x)$ ?

See all solutions

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