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Chapter 8: Q. 31 (page 680)

Find the Maclaurin series for the specified function:

cosx.

Short Answer

Expert verified

The Maclaurin series is,

f(x)=鈭�k=0鈭�(-1)k12k!x2k.

Step by step solution

01

Step 1. Given Information.

The function is,

cosx.

02

Step 2. Formulae the Maclaurin series.

Let f(x)=cosx.

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

localid="1649791589053" f(x)=f(0)+f'(0)x+f''(0)2!x2+f'''(0)3!x3+f''''(0)4!x4+.....

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

f(x)=鈭�n=0鈭�fn(0)n!xn

03

Step 3. Finding the Maclaurin series.

The table of the Maclaurin series for the function is,

The Maclaurin series for the functionf(x)=cosxis,

f(x)=1+0x+(-1)2!x2+03!x3+14!x4+05!x5+-16!x6+.....

Or, we can write it as,

f(x)=鈭�k=0n(-1)k12k!x2k.

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Most popular questions from this chapter

Let 鈭�k=0鈭�鈥�akxkbe a power series in x with an interval of convergence[-2,2). What is the radius of convergence of the power series 鈭�k=0鈭�鈥�ak(x鈭�3)k? Justify your answer.

Let ak鈮�0for each value of k, and let 鈭�k=0鈭�鈥�akxkbe a power series in xwith a positive and finite radius of convergence p. What is the radius of convergence of the power series鈭�k=0鈭�鈥�1akxk?

Find the interval of convergence for power series:鈭�k=0鈭�-1k2k!x2k.

Find the interval of convergence for each power series in Exercises 21鈥�48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

鈭�k=1鈭�2k+1kxk

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

ex
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