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Chapter 8: Q2E (page 433)

In problems 1-6, determine the convergence set of the given power series.

鈭�n=0鈭�3nn!xn

Short Answer

Expert verified

The set is, x鈭�(鈭�鈭�,鈭�).

Step by step solution

01

Step 1:To Find the Radius of convergence

Using the ratio test to determine the radius of convergence.

limn鈫�鈭�|anan+1|=limn鈫�鈭�|3nn!3n+1n+1!|=limn鈫�鈭�|3n3n+1脳(n+1)!n!|=limn鈫�鈭�|(n+1)n!n!脳3n3脳3n|=limn鈫�鈭�|n+13|=鈭�

The radius of convergence is 鈭�, therefore the series is convergent over the complete real line.

|x|<鈭�

02

Step 2:Find the set of convergence

The convergent set for the given power series is

x鈭�(鈭�鈭�,鈭�)

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Most popular questions from this chapter

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

Question: In Problems 1鈥�10, determine all the singular points of the given differential equation.

1. (x+1)y"-x2y'+3y = 0

Question: In Problems 29鈥�34, determine the Taylor series about the point x0for the given functions and values of x0.

34. f(x)=x,x0=1

In Problems 13-19,find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

y''-(cosx)y'-y=0y(蟺/2)=1,y'(蟺/2)=1

(a) Use (20) to show that the general solution of the differential equation \(xy'' + \lambda y = 0\) on the interval \((0,\infty )\) is\(y = {c_1}\sqrt x {J_1}\left( {2\sqrt {\lambda x} } \right) + {c_2}\sqrt x {Y_1}\left( {2\sqrt {\lambda x} } \right)\).

(b) Verify by direct substitution that \(y = \sqrt x {J_1}\left( {2\sqrt {\lambda x} } \right)\)is a particular solution of the DE in the case \(\lambda = 1\).
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