Maclaurin Series Informal Proof

Most students will encounter the Maclaurin Series (also known as the Taylor’s Series centered at zero) when they are studying JC H2 or College Maths. The formula looks pretty intimidating at the start:

$\displaystyle \boxed{f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\cdots+\frac{x^n}{n!}f^{(n)}(0)+\cdots}$

How on earth does one come up with that formula?

However, it turns out it is not that hard to prove the Maclaurin Series informally, or at least to derive the above formula. (The hard part is related to rigorous proof of convergence, etc.)

The idea is to approximate a function by a power series (a kind of infinite polynomial) and then find out what are the coefficients.

So, we assume we can write the function as such:

$\boxed{f(x)=f_0+xf_1+x^2f_2+x^3f_3+\cdots+x^nf_n+\cdots\;(\dagger)}$, where $f_i$ are the coefficients of the polynomial (to be determined).

We also assume that the above equation holds for all $x$.

Then, letting $x=0$, we get $f(0)=f_0$. We have just found the first coefficient!

Next, we differentiate the equation $(\dagger)$ to get:

$f'(x)=f_1+2xf_2+3x^2f_3+\cdots+nx^{n-1}f_n+\cdots$

Letting $x=0$ again, we get: $f'(0)=f_1$.

Now, differentiating the above equation one more time gives us:

$f''(x)=2f_2+6xf_3+\cdots+n(n-1)x^{n-2}f_n+\cdots$

Letting $x=0$ gives $\displaystyle f_2=\frac{f''(0)}{2}$.

Keep on differentiating, and we will see that $\displaystyle f_n=\frac{f^{(n)}(0)}{n!}$.

This is how we get the Maclaurin Series: 🙂

$\displaystyle \boxed{f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\cdots+\frac{x^n}{n!}f^{(n)}(0)+\cdots}$

Excellent Video by Khan Academy

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