## Monday, 3 February 2020

### Taylor and Maclaurin series

In section 1.10 Carroll Taylor-expands a Lagrangian. So I had a look at Taylor series. There is a very good article on Wikipedia about them.

In particular I looked at the Taylor series of 1/(1-x), polynomials especially 3rd order, sines and 'other functions' which states that they often converge to their Taylor series expansion which is a technique often used in physics.

I was able to proved that a 3rd order polynomial is its Taylor series, that is:$$k_0+k_1x+k_2x^2+k_3x^3=\sum_{i=0}^{i=3}\left(\left(\sum_{n=i}^{n=3}{\frac{n!}{\left(n-i\right)!}k_na^{n-i}}\right)\left(\sum_{j=0}^{j=i}\frac{x^{i-j}\left(-a\right)^j}{j!\left(i-j\right)!}\right)\right)$$The bit on the right is the Taylor series (after a bit of work) and there are many terms like  $a^ix^j$ which amazingly cancel each other out. I was not able to prove that any polynomial is its Taylor series as Wikipedia states.

The Taylor series for the sine function is $$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\ldots$$which is Madhava's sine series, the well known formula for $\sin{x}$. It was discovered in the west by Isaac Newton (1670) and Wilhelm Leibniz (1676). Taylor prospered in the 1700's so perhaps he found it independently. But since Newton and Leibniz independently invented calculus and argued about who was first, they probably new quite a bit about Taylor and McLaurin series. However they were all preceded by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics, who proved it as recorded in 1530 in the Indian text the Yuktibhāṣā. The picture shows the sine Taylor series for terms up to ${x^15}/{15!}$. Amazingly accurate!
Read it all in Commentary 1.10 Taylor and Maclaurin series.pdf (4 pages)