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Chapter 8: Q. 9 (page 700)

If fis a function such that f(0)=2and f'(x)=-3f(x)for every value of x, find the Maclaurin series for f.

Short Answer

Expert verified

The Maclaurin series for the function f(x)is.

f(x)=21-3x+9x22!-27x33!+鈥�

Or, it can be written as

f(x)=2鈭�k=0鈭�(-1)k3kxkk!

Step by step solution

01

given information

Let us consider the function such that that f(0)=1 and f'(x)=-3f(x)

02

calculation

The general formula to calculate the Maclaurin series for the function is:

f(x)=f(0)+f'(0)x+f''(0)2!x2+f''(0)3!x3+鈥�

As, localid="1650381164301" f(0)=2

f'(0)=-3f(0)=-3路2=-6

So,

f''(0)=-3f'(0)=-3路(-6)=18

And

fm(0)=-3f''(0)=-3路18=-54

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Most popular questions from this chapter

In Exercises 23鈥�32 we ask you to give Lagrange鈥檚 form for the corresponding remainder, R4(x)

sinx

Prove that if the power series 鈭�k=0鈭�鈥�akxkand 鈭�k=0鈭�鈥�akx2khave the same radius of convergence p, then pis 0,1, or infinite.

Let f be a twice-differentiable function at a point x0. Explain why the sum

f(x)+f'(x)(x-x0)+f''(x)2!(x-x0)2

is not the second-order Taylor polynomial for f at x0.

The second-order differential equation

x2y''+xy'+x2-p2=0

where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by Jp(x).It may be shown that Jp(x) is given by the following power series in x:

Jp(x)=鈭�k=0鈭�(-1)kk!(k+p)!22k+px2k+p

What is the interval of convergence forJp(x) where p is a non-negative integer

Prove that if (-p,p]is the interval of convergence for the series 鈭�k=0鈭�鈥�akxk, then the series converges conditionally at p.

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