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Chapter 8: Q.33 (page 680)

Find the Maclaurin series for the specified function:

sinx.

Short Answer

Expert verified

The Maclaurin series is,

f(x)=鈭�k=0鈭�(-1)k(2k+1)!x2k+1.

Step by step solution

01

Step 1. Given Information.

The function is,

sinx.

02

Step 2. Formulae for the Maclaurin series.

Let f(x)=sinx.

Since, for any function fwith derivatives of all orders at the point x0=0then the Maclaurin series is,

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

Or, we can write the general form Maclaurin series of the function fis,

localid="1649653268867">f(x)=鈭�n=0鈭�fn(0)n!xn.

03

Step 3. Finding the Maclaurin series.

So, let us first construct the table of the Maclaurin series for the function f(x)=sinx,

Therefore the Maclaurin series for the functionf(x)=sinxis,

0+1.x+02!x2+(-1)3!x3+0x4+15!x5+06!x6+......

Or, we can write as,

f(x)=鈭�k=0鈭�(-1)k(2k+1)!x2k+1.

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Most popular questions from this chapter

What is Lagrange鈥檚 form for the remainder? Why is Lagrange鈥檚 form usually more useful for analyzing the remainder than the definition of the remainder or the integral provided by Taylor theorem?

Show that the power series 鈭�k=1鈭�鈥�(鈭�1)kk2xkconverges absolutely when x=1and when x=-1. What does this behavior tell you about the interval of convergence for the series?

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of x0. Note: These are the same functions and values as in Exercises 41鈥�48.

54.x3,1

The second-order differential equation

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

where p is a nonnegative 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

Graph the first four terms in the sequence of partial sums of J1(x).

What is x0if the interval of convergence for the power series 鈭�k=0鈭�鈥�akx鈭�x0kis(2,10]?
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