Home > Calculus > Constructing a Maclaurin Series

How do you find the Maclaurin series of #f(x)=e^x# ?

Answer 1

Evelyn Agard

You have:

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Elijah Abram

To find the Maclaurin series of ( f(x) = e^x ), you can use the definition of the Maclaurin series, which is given by:

[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n ]

For ( f(x) = e^x ), the derivatives are:

[ f'(x) = e^x ] [ f''(x) = e^x ] [ f'''(x) = e^x ] [ \text{and so on...} ]

Evaluating these derivatives at ( x = 0 ), we find that:

[ f(0) = e^0 = 1 ] [ f'(0) = e^0 = 1 ] [ f''(0) = e^0 = 1 ] [ f'''(0) = e^0 = 1 ] [ \text{and so on...} ]

Thus, the Maclaurin series for ( e^x ) is:

[ e^x = \sum_{n=0}^{\infty} \frac{1}{n!}x^n ]

Or more compactly:

[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} ]

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.