Ramanujan-type series for [one divided by pi] with quadratic irrationals : a thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand
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Date
2012
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Massey University
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Abstract
In 1914, Ramanujan discovered 17 series for 1= , 16 are rational and one is irrational.
They are classi ed into four groups depending on a variable ` called the level, where
` = 1, 2, 3 and 4. Since then, a total of 36 rational series have been found for these
levels. In addition, 57 series have been found for other levels. Moreover, 14 irrational
series for 1= were found. This thesis will classify the series that involve quadratic
irrationals for the levels ` 2 f1; 2; 3; 4g. A total of 90 series are given, 76 of which are
believed to be new. These series were discovered by numerical experimentations using
the mathematical software tool \Maple" and they will be listed in tables at the end of
this thesis.
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Keywords
Ramanujan-type series, Pi, Quadratic irrationals