**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).

Given, f(x)=11+x2 Comparing this with a1-r, we get a=1, r=-x2

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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