Question

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = e−3x

Answer 1

The equation of
`f(x) = e^(-3\cdot x)` by Maclaurin series is
`f(x) = \Sigma\limits_(i=0)^(\infty) ((-3\cdot x)^(i))/(i!)`.

Explanation:

The Maclaurin series for
`f(x)` is defined by the following formula:

`f(x) = \Sigma\limits_(i = 0)^(\infty) (f^((i))(0))/(i!) \cdot x^(i)` (1)

Where
`f^((i))` is the i-th derivative of the function.

If
`f(x) = e^(-3\cdot x)`, then the formula of the i-th derivative of the function is:

`f^((i)) = (-3)^(i)\cdot e^(-3\cdot x)` (2)

Then,

`f^((i))(0) = (-3)^(i)` (2b)

Lastly, the equation of the trascendental function by Maclaurin series is:

`f(x) = \Sigma\limits_(i=0)^(\infty) ((-3)^(i)\cdot x^(i))/(i!)`

`f(x) = \Sigma\limits_(i=0)^(\infty) ((-3\cdot x)^(i))/(i!)` (3)