Does 0.999… = 1? And Are Divergent Series the Invention of the Devil?

Which refers to a saying by Niels Henrik Abel

The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes …

This reminded me about a fuss in the blogosphere a few months ago:

• Phil Plait, the “Bad Astronomer” ventured into purely mathematical territory: When Infinity Is Actually a Small, Negative Fraction

• Which provoked Bad Math from the Bad Astronomer

  The important thing to note here is that we are not saying that the Cesaro sum is equal to the series. We’re saying that there’s a way of assigning a measure to the series.

  And there is the first huge, gaping, glaring problem with the video. They assert that the Cesaro sum of a series is equal to the series, which isn’t true.

• and If it’s hocus pocus then it’s not math

Some other comments:

• Infinite series: not quite as weird as some would say
• Correction: Does 1+2+3+4+ . . . =-1/12? Absolutely Not
• So What Does 1+2+3+4+… Equal? We Give You the Answer
• Infinity or -1/12?

While I majored in physics at Carleton I also took more math than was required, enough to appreciate limits and convergence, and to see the dangers in saying that a non-convergent series is equal to anything. But it is kind of fun.