General Mathematics, Vol. 10,nr. 1-2, 2002

       Alexandru Lupas and Luciana Lupas - Rational sequences converging to π

Abstract:

Our aim is to give sequences $ \sir{\bq_n} $ and $ \sir{\BQ_n} $
of rational numbers such that\hsp $ \bq_n < \bq_{n+1}< \pi <
\BQ_{k+1}<\BQ_k\; ,\hsp n,k\in \NN\; . $ It is shown that there
exists positive constants $ C_1,C_2 $ such that for $ n $ large ,
$ \ds |\bq_n -\pi|< \ds \frac{C_1}{ {2}^{5n}} $ and $ \ds
|\BQ_n-\pi|< \ds \frac{C_2}{n\cdot {2}^{5n}} $ . Let us note that
both sequences are constructed by means of the same three-term
recurrence relation. Likewise, two series for π are given.

Full text of the article:

• Postscript file
• PDF file