Answer :
The maclaurin expansion for function eˣ is 1 + x + x²/2 + x³/6 + x⁴/24 + .....
A function's expansion series, which provides the function's derivatives' sum, is known as a Maclaurin series.
Series [f, x, 0, n] can be used to find a function's Maclaurin series up to order n.
When x = 0, it is a specific example of the Taylor series. The series of Maclaurin is provided by
[tex]f(x) = f(x_0) + f(x_0)(x - x_0) + \frac{f"(x_0)}{2!}(x - x_0)^2 + . . .[/tex]
The formula for maclurin series is :
[tex]f(x) = \sum^{\infty}_{n=0} \frac{f^n (x_0)}{n!} (x-x_0)[/tex]
Where,
f(xo), f’(xo), f’‘(xo)……. are the successive differentials when xo = 0.
Maclaurin expansion for the function eˣ
Firstly , We will derivate the function
f(x) = eˣ
=> f'(x) = e⁰ = 1
=> f''(x) = e⁰ = 1
=> f'''(x) = e⁰ = 1
and so on
Upon evaluation of all derivatives at x = 0, we obtain the value 1.
Moreover, f(0)=1, so we can deduce that the expansion of the Maclaurin Series as
=> eˣ = 1 + x + x²/2 + x³/6 + x⁴/24 + .....
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