Find the Maclaurin series of $ f(z) = \frac{z^2}{1 - 9z^2} $ and $ g(z) = z^2 e^{-z^3} + 2z - 3 $, and find the radius of convergence of each.

Given the function . We need to find the Maclaurin series.We can use .Replace in place of z, we…

Answered: Find the Maclaurin series of f(z) =…

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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Find the Maclaurin series of $ f(z) = \frac{z^2}{1 - 9z^2} $ and $ g(z) = z^2 e^{-z^3} + 2z - 3 $, and find the radius of convergence of each.

Expert Solution

Step 1: Finding the Maclaurin series for f(z)

Given the function $f open parentheses z close parentheses equals fraction numerator z squared over denominator 1 minus 9 z squared end fraction$ . We need to find the Maclaurin series.

We can use $1 plus z plus z squared plus z cubed plus horizontal ellipsis equals fraction numerator 1 over denominator 1 minus z end fraction comma space open vertical bar z close vertical bar less than 1$ .

Replace $9 z squared$ in place of $z$ , we get

$1 plus open parentheses 9 z squared close parentheses plus open parentheses 9 z squared close parentheses squared plus open parentheses 9 z squared close parentheses cubed plus horizontal ellipsis equals fraction numerator 1 over denominator 1 minus open parentheses 9 z squared close parentheses end fraction comma space open vertical bar 9 z squared close vertical bar less than 1 1 plus 9 z squared plus 81 z to the power of 4 plus 729 z to the power of 6 plus horizontal ellipsis equals fraction numerator 1 over denominator 1 minus open parentheses 9 z squared close parentheses end fraction$

Multiplying both sides by $z^{2}$ ,

$z squared plus 9 z to the power of 4 plus 81 z to the power of 6 plus 729 z to the power of 8 plus horizontal ellipsis equals fraction numerator z squared over denominator 1 minus open parentheses 9 z squared close parentheses end fraction$

Thus, $fraction numerator z squared over denominator 1 minus open parentheses 9 z squared close parentheses end fraction equals sum from n equals 0 to infinity of 9 to the power of n z to the power of 2 n plus 2 end exponent$ .

The radius of convergence is

$open vertical bar 9 z squared close vertical bar less than 1 rightwards double arrow open vertical bar z squared close vertical bar less than 1 over 9 rightwards double arrow open vertical bar z close vertical bar less than 1 third.$

Thus, the radius of convergence is $R = \frac{1}{3} .$