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Exercise J : by J(*) 17(1 + x"). Show that for every x E R there exists & > 0 such that ƒ is represented by a power series convergent in (x – 6, x + 8), although the power series at the origin does not converge on the entire line. |

Exercise J : by J(*) 17(1 + x"). Show that for every x E R there exists & > 0 such that ƒ is represented by a power series convergent in (x – 6, x + 8), although the power series at the origin does not converge on the entire line. |

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Exercise 3.1.** Define \( f : \mathbb{R} \rightarrow \mathbb{R} \) by \( f(x) = 1/(1 + x^2) \). Show that for every \( x \in \mathbb{R} \) there exists \( \delta > 0 \) such that \( f \) is represented by a power series convergent in \( (x-\delta, x+\delta) \), although the power series at the origin does not converge on the entire line. *Hint:* This is an example of an application of complex analysis to real analysis. The function in the exercise is thus real-analytic on all of \( \mathbb{R} \) but should not really be called real-entire (at least in our opinion; some authors differ on this point).
Transcribed Image Text:**Exercise 3.1.** Define \( f : \mathbb{R} \rightarrow \mathbb{R} \) by \( f(x) = 1/(1 + x^2) \). Show that for every \( x \in \mathbb{R} \) there exists \( \delta > 0 \) such that \( f \) is represented by a power series convergent in \( (x-\delta, x+\delta) \), although the power series at the origin does not converge on the entire line. *Hint:* This is an example of an application of complex analysis to real analysis. The function in the exercise is thus real-analytic on all of \( \mathbb{R} \) but should not really be called real-entire (at least in our opinion; some authors differ on this point).
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Step 1

Given, f(x)=11+x2

Comparing this with a1-r, we get a=1, r=-x2

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