The development of the theory of summable divergent series from to | SpringerLinkIt is a divergent series , meaning that it lacks a sum in the usual sense. Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value". Variations of this idea, called the Eilenberg—Mazur swindle , are sometimes used in knot theory and algebra. Treating Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value:. The above manipulations do not consider what the sum of a series actually means and how said algebraic methods can be applied to divergent geometric series. Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions:.
Theory And Application Of Infinite Series
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