# How do you find the Maclaurin series for #e^x#?

which is most likely among the most significant power series in mathematics!

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To find the Maclaurin series for ( e^x ), you can use the definition of the Maclaurin series, which is:

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

For ( e^x ), the derivatives are:

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

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

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

Substituting these values into the Maclaurin series definition, we get:

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

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

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

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