# On the origins of Taylor/Maclaurin Series

Many a times it is not discussed as to How the Taylor/Maclaurin series came to be in its current form. This short snippet is all about it.

Let us assume that some function $f(x)$ can be written as a power series expansion. i.e

$f(x) = a_0 + a_1 x + a_2 x^2 + \hdots$.

We are left with the task of finding out the coefficients of the power series expansion.

Substitution x = 0, we obtain the value of $a_0$.

$a_0 = f(0)$.

Lets differentiate $f(x)$ wrt x.

$\frac{d}{dx} f(x) = a_1 + 2a_2 x + \hdots$

Evaluating at x =0 , we get

$\frac{d}{dx} f(0) = a_1$

And likewise:

$\frac{d^2}{dx^2} f(0) = 2.1.a_2 = 2! \space a_2$

$\frac{d^3}{dx^3} f(0) = 3.2.1.a_3 = 3! \space a_3$

$\vdots$

$\frac{d^n}{dx^n} f(0) = n.n-1...3.2.1.a_n = n! a_n$

That’s it we have found all the coefficient values, the only thing left to do is to plug it back into the power series expression:

$f(x) = f(0) + \frac{d}{dx}f(0) \frac{x}{1!} + \frac{d^2}{dx^2}f(0) \frac{x^2}{2!} + \frac{d^3}{dx^3} f(0) \frac{x^3}{3!} \hdots$.

The above series expanded about the point x = 0 is called as the ‘Maclaurin Series’. The same underlying principle can be extended for expanding about any other point as well i.e ‘Taylor Series’.

# Legendre Differential Equation(#2): A friendly introduction

Now there is something about the Legendre differential equation that drove me crazy. What is up with the l(l+1) !!!

$(1-x^2)y^{''} -2xy^{'} + l(l+1)y = 0$

To understand why let’s take this form of the LDE and arrive at the above:

$(1-x^2)y^{''} -2xy^{'} + \lambda y = 0$

$y = \sum\limits_{n=0}^{\infty} a_n x^n$

If we do a power series expansion and following the same steps as the previous post, we end up with the following recursion relation.

$(n+2)(n+1)a_{n+2} = (\lambda -n(n+1))a_n$

or

$a_{n+2} = a_n \frac{\lambda - n(n+1)}{(n+1)(n+2)}$

Here’s the deal: We want a convergent solution for our differential solution. This means that as $n \rightarrow l , a_{n+2} \rightarrow 0$.

Hence we obtain that

$\lambda = l(l+1)$