• Exercise 3.1. Define f : R → R by f(x)1/(1 + x2). Show that forevery x ER there exists & > 0 such that f is represented by a powerseriesconvergent in (x – 6, x+8), although the power series at the origin does notconverge on the entire line.

Given a real-valued function f defined as f(x)=11+x2 Rewrite this function as f(x)=11-(ix)2 Consider…

Answered: • Exercise 3.1. Define f : R → R by…

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 : R → R by f(x)
1/(1 + x2). Show that for
every x ER there exists & > 0 such that f 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.

Expert Solution

Step 1

Given a real-valued function $f$ defined as

$f (x) = \frac{1}{1 + x^{2}}$

Rewrite this function as

$f (x) = \frac{1}{1 - {(i x)}^{2}}$

Consider the power series

$\sum_{n = 0}^{\infty} {(i x)}^{2 n} = 1 + {(ix)}^{2} + {(ix)}^{4} + {(ix)}^{6} + {(ix)}^{8} + \dots \dots$

This is a geometric series with first term $1$ and common ratio ${(ix)}^{2}$ and converges for $| x | < 1$

Step 2

The sum of this series

$\sum_{n = 0}^{\infty} {(i x)}^{2 n} = 1 + {(ix)}^{2} + {(ix)}^{4} + {(ix)}^{6} + {(ix)}^{8} + \dots \dots$

is given as

$\sum_{n = 0}^{\infty} {(i x)}^{2 n} = \frac{1}{1 - {(ix)}^{2}}$

$\sum_{n = 0}^{\infty} {(i x)}^{2 n} = \frac{1}{1 - i^{2} x^{2}}$

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• Exercise 3.1. Define f : R → R by f(x) 1/(1 + x2). Show that for every x ER there exists & > 0 such that f 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.