<P> 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 . </P> <P> The above manipulation might be called on to produce − 1 outside 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 . These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as 0.111...and most notably 0.999.... The arguments are ultimately justified for these convergent series, implying that 0.111...= 1 / 9 and 0.999...= 1, but the underlying proofs demand careful thinking about the interpretation of endless sums . </P> <P> It is also possible to view this series as convergent in a number system different from the real numbers, namely, the 2 - adic numbers . As a series of 2 - adic numbers this series converges to the same sum, − 1, as was derived above by analytic continuation . </P>

Sum of 1 + 2 + 4 + 8