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Chapter 8: Q-14E (page 434)

Question:In Problem find the first three nonzero terms in the power series expansion for the product f(x)g(x).

Short Answer

Expert verified

The required product is, f(x).g(x)=1.

Step by step solution

01

Given power series expansion

The series expansion of the given series is given by

02

Find the product of the power series expansion

Multiplying both the series

There is only one non-zero term in the series expansion, that is f(x).g(x)=1.

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Most popular questions from this chapter

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

x2y"+2xy'-3y=0

Question: Let

Show that fn(0)=0for n=0,1,2....and hence that the Maclaurin series for f(x)is 0+0+0+...., which converges for all xbut 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.

In Problems 15-17, solve the given initial value problem t2x"-12x=0. x(1)=3 and x'(1)=5.

In Problems \(5 - 14\) solve the given linear system.

\({\bf{X'}} = \left( {\begin{array}{*{20}{c}}{{\rm{ 0 2 1}}}\\{1{\rm{ }}1{\rm{ }} - 2}\\{2{\rm{ }}2{\rm{ }} - 1}\end{array}} \right){\bf{X}}\)

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

(1-x2) y"-y'+y=tan x
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