Taylor Series

My diurnal age today is 24491 and this number features in OEIS A077946:  

A077946

Expansion of ${\displaystyle \frac{1}{1-x-2x^2-2x^3}}$

The number arises as a coefficient of $x$ in the Taylor Series expansion of the function at $x=0$. The coefficients listed up to 24491 are:

1, 1, 3, 7, 15, 35, 79, 179, 407, 923, 2095, 4755, 10791, 24491

This means that the function can be expressed as:

$1+x+3x^2+7x^3+15x^4+35x^5+\dots$

The Taylor Series for any continuous function $f(x)$at a point $x=a$ is given by the following expression:$$f(x)=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{″}(a)}{a!}(x-a{)}^2+\frac{f^{‴}(a)}{3!}(x-a{)}^3+\dots$$A Maclaurin Series is a Taylor Series where $a=0$.

Here is a permalink that will display these results using SageMathCell. I have a later post about Taylor Series that I created on April 27th 2018.

REFURBISHED on Saturday November 26th 2022