Please explain what an infinitesimal is, without using circular logic, and so that things such as a sum of a series will make sense given the definition.
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Anonymous2010-01-03 0:16
Why? Infinitesimals were a way to explain calculus before it was put on more rigorous foundations. They aren't necessary for understanding sequences and series.
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Anonymous2010-01-03 0:35
An infinitesimal is defined as a number such that, if multiplied by any natural number, is never superior to one (or any number, really).
In conventional mathematics, 0 is an infinitesimal, and is the only one. You can also use hyperreal sets and such for more infinitesimals.
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Anonymous2010-01-03 0:52
>Please explain what an infinitesimal is
There was never a water-tight mathematical definition of an infinitesimal (as used by Newton, etc); the notion made sense to mathematicians only on an intuitive level. In the 19th century the epsilon-delta definition of a limit was invented, which made sense mathematically, and could be used in calculus instead of infinitesimals.