1 + 2 + 4 + 8 + · · ·
978-613-1-13762-4
6131137625
108
2010-11-13
39.00 €
eng
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In a useful sense, s = ∞ is a root of the equation s = 1 + 2s. (For example, ∞ is one of the two fixed points of the Möbius transformation z → 1 + 2z on the Riemann sphere.) If some summation method is known to return an ordinary number for s, i.e. not ∞, then it is easily determined. In this case s may be subtracted from both sides of the equation, yielding 0 = 1 + s, so s = −1. The above manipulation might be called on to produce −1 outside of the context of a sufficiently powerful summation procedure. For the most well-known and straightforward sum concepts, including the fundamental convergent one, it is absurd that a series of positive terms could have a negative value. A similar phenomenon occurs with the divergent geometric series 1 − 1 + 1 − 1 + · · ·, where a series of integers appears to have the non-integer sum 1⁄2.
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Mathematics
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