Abstract
Grandi’s paradox, which was posed for a real function of the form 1/(1+ x), has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions. Furthermore, a new technique for improving the convergence characteristics of power series with alternating signs is introduced. The technique works by successively averaging a series at different orders of truncation. A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the function 1/(1+ ex ) . Grandi-type complex valued functions such as 1/(i + x) are expressed as consistently-truncated and convergence-improved forms and Fagnano’s formula is established from the series expansions of these functions. A Grandi-type general complex valued function is introduced and expanded to a consistently truncated and successively averaged series. Finally, an unorthodox application of the successive averaging method to polynomials is presented.
Highlights
Luigi Guido Grandi (1671-1742) is known due to his book entitled Quadratura in short
Grandi’s paradox, which was posed for a real function of the form 1 (1+ x), has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions
A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the
Summary
Luigi Guido Grandi (1671-1742) is known due to his book entitled Quadratura in short. The book does not contain much original work except for two particular items; namely, the construction of a curve that has become known as the Witch of Agnesi and the identification of a paradox originating from the series expansion of 1 (1+ x). Consider the series expansion of 1 (1+ x) , which may be obtained by performing a simple division or expanding into a Maclaurin series. Before proceeding to the resolution of the paradox the series expansion in (1) is rendered convergent for x > 1 by a simple manipulation as follows.
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