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Chapter 8: Q.8 (page 659)

If fis a function such that f(0)=1and f'(x)=f(x)for every value of x, find the Maclaurin series for f.

Short Answer

Expert verified

The Maclaurin series for the function is:

f(x)=1+x+x22!+x33!+鈥�

Or, it can be written as

ex=鈭�k=0鈭�1k!xk

Step by step solution

01

Given Information

Given equations :

f(0)=1f'(x)=f(x)

02

Finding the Maclaurin series for f

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

So, the function must be f(x)=ex

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�

As role="math" localid="1650439558986" f(0)=1, therefore role="math" localid="1650439565403" f'(0),f''(0)andf''(0)are1

So, The Maclaurin series for the function is:

role="math" localid="1650439570178" f(x)=1+x+x22!+x33!+鈥�鈬�ex=鈭�k=0鈭�1k!xk

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