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Q-37E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 435

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Question: Let
Show that fⁿ(0)=0 for n=0,1,2.... and hence that the Maclaurin series for f(x) is 0+0+0+.... , which converges for all x but is equal to f(x) only when x=0 . This is an example of a function possessing derivatives of all orders (at x₀=0 ), whose Taylor series converges, but the Taylor series (about x₀ =0) does not converge to the original function! Consequently, this function is not analytic at x=0.

Function is not analytic at x=0 .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Taylor series

For a function f(x) the Taylor series expansion about a point x₀ is given by, $f (x - x_{0}) = f (x_{0}) + f' (x_{0}) . (x - x_{0}) + f " (x_{0})$ . $\frac{{(x - x_{0})}^{2}}{2!} + f "' (x_{0}) . \frac{{(x - x_{0})}^{3}}{3!} + . . . .$

Step 2: The derivatives of f

Let $x \neq 0$ , then f(x)=e $^{- \frac{1}{x^{2}}}$ .

For , the derivatives of f are:

f'(x)=e

f''(x)=e $(\frac{2}{x^{6}} - \frac{6}{x^{4}})$

f'''(x)=e $(\frac{- 12}{x^{7}} + \frac{24}{x^{5}} + \frac{4}{x^{9}} - \frac{6}{x^{4}})$

Note that f⁽ⁿ⁾(x) will be of the form e p(x), where p(x) is some polynomial.

From the definition of f it follows that, f(0)=0 .

Step 3: To calculate the derivative at x=0

We need to calculate the derivative at x=0 by definition,

Most popular questions for Math Textbooks

In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

$y'' - e^{2 x} y' + (cosx) y = 0 y (0) = - 1, y' (0) = 1$

In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

$y'' - (cosx) y' - y = 0 y (π / 2) = 1, y' (π / 2) = 1$

In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

$y'' - (sinx) y = 0; y (π) = 1, y' (π) = 0$

In Problems 1-10, use the substitution y=x^r to find a general solution to the given equation for x>0.

x³y"'+3x²y"+5xy'-5y=0

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four non-zero terms in a power series expansion about’s x=0 of a general solution to the given differential equation.

w'+xw=e^x

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