power 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’.

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.

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

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

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

$\lambda = l(l+1)$

Ecstasy Shots

A poetry of logical ideas

On the origins of Taylor/Maclaurin Series

Legendre Differential Equation(#2): A friendly introduction