the Taylor's expansion of $\log \frac{\sin(x)}x$

Question

How does one write the Taylor expansion of $\log\big(\frac{\sin(x)}x\big)$? It is not defined on $x=0$.

Answer 1

If you are not looking for the infinite series, since $\forall x$, $\frac{\sin(x)} x <1$ and that $x=0$ is a removable singularity, let $a=\frac{\sin(x)} x -1$ and write $$\log \left(\frac{\sin(x)}x\right)=\log(1+a)$$ Now expand as usual $$\log(1+a)=a-\frac{a^2}{2}+\frac{a^3}{3}-\frac{a^4}{4}+\frac{a^5}{5}-\frac{a^6}{6}+\frac{a^7}{7}+O\left(a^8\right)\tag 1$$ and now $$a=\frac{\sin(x)} x -1=-\frac{x^2}{6}+\frac{x^4}{120}-\frac{x^6}{5040}+\frac{x^8}{362880}+O\left(x^{10}\right)$$ Now, for the computation of $a^n$, use th binomial expansion stopping a soon as you encounter power of $x$ gerater that $8$ (for axample). This would give $$a^2=\frac{x^4}{36}-\frac{x^6}{360}+\frac{41 x^8}{302400}+O\left(x^{10}\right)$$ $$a^3=-\frac{x^6}{216}+\frac{x^8}{1440}+O\left(x^{10}\right)$$ and so on.

Replacing in $(1)$ this would give $$\log \left(\frac{\sin(x)}x\right)=-\frac{x^2}{6}-\frac{x^4}{180}-\frac{x^6}{2835}-\frac{x^8}{37800}+O\left(x^{10}\right)$$

Now, if you are curious, ask $OEIS$ for sequence $\{6,180,2835,37800\}$ and you will find sequence $A046989$ which will show you the beautiful formula given by @user76284.

Answer 2

According to OEIS,

$$ \log \frac{x}{\sin x} = \sum_{n > 0} x^{2n} \frac{2^{2n-1} (-1)^{n+1} B(2n)}{n (2n)!} $$

where $B$ is the Bernoulli number. Note that the singularity at $x=0$ is removable.

Answer 3

Euler product formula for sine:

$$\frac{\sin x}{x}=\prod\limits_{k=1}^\infty\left(1-\left(\frac{x}{\pi k}\right)^2\right)$$

You see: $~\ln\frac{\sin x}{x}~$ is defined everywhere in arbitrary proximity of $~x=0~$, the result is $~0~$.

With the help of

$$-\ln\prod\limits_{v=1}^n\left(1-za_v\right)=\sum\limits_{k=1}^\infty\frac{z^k}{k}\sum\limits_{v=1}^n a_v^k$$

we can set $\displaystyle~a_k:=\frac{1}{(\pi k)^2}~$ and get:

$$\ln\frac{\sin x}{x}=-\sum\limits_{k=1}^\infty\frac{x^{2k}}{k}\frac{\zeta(2k)}{\pi^{2k}}$$

Note: $~~\zeta(s)~$ is called the Riemann Zeta function . $~~~\frac{\zeta(2k)}{\pi^{2k}}~$ are rational numbers.