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Chapter 8: Q . 8 (page 700)

If f is a function such thatf(0)=1andf'(x)=f(x)for every value of x, find the Maclaurin series for f.

Short Answer

Expert verified

So, The Maclaurin series for the function is:

f(x)=1+x+x22!+x33!+鈰�

Or, it can be written as

ex=鈭�i=0鈭�1k!xk

Step by step solution

01

given information

Let us consider the function such that that f(0)=Iandf'(x)=f(x)

So, the function must bef(x)=ex

02

calculation

Since, 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�

Asf(0)=1, therefore f'(0),f''(0)andf''(0)are 1 .

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Most popular questions from this chapter

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.

What is Taylor鈥檚 Theorem?

What is x0if the power series 鈭�k=0鈭�鈥�akx鈭�x0kconverges conditionally at both x=-4and x=8.

Complete Example 4 by showing that the power series 鈭�k=0鈭�鈥�(鈭�1)kk3k+1(2x鈭�3)kdiverges when x=2.

What is x0if (p,q)is the interval of convergence for the power series 鈭�k=0鈭�鈥�akx鈭�x0k?
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