# Number 8 Is Interesting

Eight is the first (after the trivial 1) cube, 2³ = 8 [Bunch]. Eight is the only cube 1 less than a square, _{-6} = -8,_{-2} = -1,_{0} = 0,_{-1} = u_{1} = u_{2} = 1,_{6} = 8 [Bunch, Roberts].

Bryan Bunch's and Joe Roberts' books list many more curiosities with 8 in a prominent role. The former is almost entirely elementary, the latter often refers to more advanced topics from college mathematics. A novel property of 8 has been recently made public by Paul and Vincent Steinfeld of Germany.

Consider the sequence

(*) |
| ... |

Steinfelds suggest an interpretation of what the terms that come after one runs out of decimal digits may look like. To boot, define

N_{n} | = ∑^{n}_{k = 0} (10 - (k + 1)) 10^{ - k} and | |

D_{n} | = ∑^{n}_{k = 0} (k + 1) 10^{ - k}. |

Then, with a suitable placement of the decimal point, the ratio _{n} = N_{n} / D_{n}

The limits lim_{n→∞}N_{n} and lim_{n→∞}D_{n} are easily computed to be

lim_{n→∞}N_{n} | = 800/81 and | |

lim_{n→∞}D_{n} | = 100/81. |

So that lim_{n→∞}R_{n} = 8. Unexpectedly, the sequence (*) converges to 8!

## References

- B. Bunch,
*The Kingdom of Infinite Number: A Field Guide*, W. H. Freeman & Co., 2000 - J. Roberts,
*Lure of the Integers*, MAA, 1992 - P. and V. Steinfeld,
__Math Bite: A Magic Eight__,*Mathematics Magazine*, Vol. 82, No. 1, Feb. 2009, p. 25

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First we are going to show that lim_{n→∞}D_{n} = 100/81. Note that the series at hand is *absolutely convergent* (it's convergent and consists of positive terms), meaning that it is possible to reshuffle the terms without affecting the limit. With the formula for the sum of the geometric series in mind, we have

∑^{∞}_{k = 0} (k + 1) 10^{ - k} | = ∑^{∞}_{k = 0} 10^{ - k} | + ∑^{∞}_{k = 1} k 10^{ - k} | |||

= ∑^{∞}_{k = 0} 10^{ - k} | + ∑^{∞}_{k = 1} 10^{ - k} | + ∑^{∞}_{k = 2} 10^{ - k} | + ∑^{∞}_{k = 3} 10^{ - k} ... | ||

= 10/9 | + 1/9 | + 10^{-1} /9 | + 10^{-2} /9 ... | ||

= 10/9 (1 | + 10^{-1} | + 10^{-2} | ...) | ||

= (10/9)² | = 100/81, |

as claimed.

For the limit lim_{n→∞}N_{n}, observe that the series is again *absolutely convergent* (the series of absolute values is convergent), and

N_{n} = 10 ∑^{∞}_{k = 0} 10^{ - k} - D_{n},

such that in the limit we get

lim_{n→∞}N_{n} | = 10×10/9 - 100/81 | |

= 900/81 - 100/81 | ||

= 800/81. |

**Note**: You may want to check that, in base b > 2, the role of 8 is taken by

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