A Maclaurin series is an infinite polynomial representation of a function $f(x)$. The reason we represent a function as a polynomial is that polynomials are easier to differentiate, integrate, and approximate numerically.
The key idea behind a Maclaurin series is to create a polynomial $P(x)$ such that at $x=0$ it matches $f(x)$ and its derivatives. That is, $P(0)=f(0)$, $P'(0)=f'(0)$, $P''(0)=f''(0)$, and similarly for all higher derivatives.
A polynomial can be defined using its coefficients, so we want to find a polynomial $P(x)=a_0+a_1x+a_2x^2+\cdots$ that satisfies the given condition.
To find the coefficients $a_n$, consider the following process:
Substitute $x=0$ into $P(x)$ to get:
$$P(0) = f(0) = a_0$$
Substitute $x=0$ into $P'(x)$ to get:
$$P'(x) = a_1 + 2a_2x + 3a_3x^2\cdots$$
$$P'(0) = f'(0)= a_1 $$
Substitute $x=0$ into $P''(x)$ to get:
$$P''(x) = 2a_2 + 6a_3x+ 12a_4x^2\cdots$$
$$P''(0) = f''(0) = 2a_2 $$
$$a_2=\frac{f''(0)}{2} $$
Substitute $x=0$ into $P'''(x)$ to get:
$$P'''(x) = 6a_3 + 24a_4x + 60a_5x^2\cdots$$
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