Find the first few terms of the power series for the

Chapter 8, Problem 15E

(choose chapter or problem)

Find the first few terms of the power series for the quotient

               

                   \(q(x)=\left(\sum_{n=0}^{\infty} \frac{1}{2^{n}} x^{n} / \sum_{n=0}^{\infty} \frac{1}{n !} x^{n}\right.)\)

by completing the following:

(a) Let \(q(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\), where the coefficients \(a_{n}\) are to be determined. Argue that \(\sum_{n=0}^{\infty} x^{n} / 2^{n}\) is the Cauchy product of \(q(x) \) and \(\sum_{n=0}^{\infty} x^{n} / n !\).

(b) Use formula (6) of the Cauchy product to deduce the equations

\(\frac{1}{2^{n}}=a_{0}, \frac{1}{2}=a_{0}+a_{1,} \frac{1}{2^{2}}=\frac{a_{1}}{2}+a_{1}+a_{2}, \frac{1}{2^{3}}=\frac{a_{1}}{6}+\frac{a_{1}}{2}+a_{3}, \ldots\)

(c) Solve the equations in part (b) to determine the constants \(a_{0}, a_{1}, a_{2}, a_{3}\).

Equation Transcription:

 

Text Transcription:

q(x)=(sum_n=0^infinity 1/2^n x^n/sum_n=0^infinity1/n! x^n)

q(x)=sum_n=0^infinity a_n x^n

a_n

sum_n=0^infinity x_n/2^n

q(x)

sum_n=0^infinity x_n/n!

1/2^0=a_0,1/2=a_0+a_1, 1/2^2= a_0/2+a_1+a_2, 1/2^3= a_0/6+ a_1/2+a_3,... .

a_0,a_1,a_2,a_3