An examination of rates of convergence of the series for $\pi$

Question

I got this topic for project "An examination of rates of convergence of the series for $\pi$". My question is which relative formulas or math knowledge I should research?

Answer 1

Just an example. Browsing thru a book on math for engineers, I found :"When using Newton's Method $x_{n+1}=x_n-f(x_n)/f'(x_n)$ to approximate $x$, where $f(x)=0$, if it is known or suspected that $f'(x)=0$, it is better to use $x_{n+1}=x_n-f(x_n)/[f'(x_n)/2].$"..... No explanation was given.

With some transcendental functions we can determine $f'(x)$ when f(x)=0 without knowing $x$. Then we can modify Newton's Method by replacing the term $f'(x_n)$ with $[f'(x_n)+f(x)]/2.$

Example: Let $f(z)=-1+\tan z $ for $|z|< \pi/2.$ Then $f'(z)=1/\cos^2 z.$ With $x=\pi/4$ we have $f(x)=0$ and $f'(x)=1/2.$ Let $x_1=0$ and $$x_{n+1}=x_n-[f(x_n)]\;/\;[(f'(x_n)+f'(x))/2]=x_n-2 f(x_n)/[f'(x_n)+1/2].$$ With $|x_n-\pi/4|=d_n,$ we have $d_{n+1}=O((d_n)^3)$, compared to Newton's Method, which only gives an error $d_n$ with $d_{n+1}=O((d_n)^2).$

You might be interested in the section "Transformation Of Slowly Converging Series" in the book Infinite Sequences And Series , by Bromwich.