Question

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

Answer 1

`e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) +... + x^n/(n!) +...`

Explanation:

The Maclaurin series is obtained by the Power Series:
`f(x) = f(0) + f'(0)x/(0!) + f''(0)x^2/(2!) + f^((3))(0)x^3/(3!) + ...`

So with `f(x)=e^x` we have
`f(x)=e^x => f(0)=1`
`f'(x)=e^x => f'(0)=1`
`f''(x)=e^x => f''(0)=1`
`f^((3))(x)=e^x => f^((3))(0)=1`
And in clearly:
`f^((n))(x)=e^x => f^((n))(0)=1`

So the Maclaurin series is:
`e^x = 1 + 1x/(0!) + 1x^2/(2!) + 1x^3/(3!) + 1x^4/(4!) +... + 1x^n/(n!) +...`
`e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) +... + x^n/(n!) +...`

Which is probably one of the most important mathematical power series !