Mathematics Rebooted by Lara Alcock

Author:Lara Alcock
Language: eng
Format: epub, pdf
Publisher: Oxford University Press
Published: 2017-08-22T14:29:00+00:00

You might like to replace r with some appropriate numbers and think about what the formula says. And you might like to know that this works for r with , too. If you’ve found these arguments fairly straightforward, work through one with , then ask yourself what happens for .

Second, the harmonic series invites comparison with similar series. For instance, consider

What do you think happens to this series? Does it converge to a finite number, or is the total infinite? I think this is not obvious. On the one hand, this is ‘like’ the harmonic series except that the denominators are of the form n2 instead of n. So maybe it diverges to infinity. On the other hand, its terms get smaller considerably faster, so maybe it behaves more like a geometric series. What do you reckon? If you find that you really don’t know, that’s good. It means that you’ve understood well enough to recognize that both outcomes are plausible.

To get some traction, a sensible strategy is to compare with known series, perhaps using the following lists. Does this help?

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