Could someone explain this contradiction to me?


Reading on with Julian Havil’s Gamma: Exploring Euler’s Constant and inspired by his discussion of the harmonic series, I come across this:

\frac{1}{1-e^x} = 1 + e^x + e^{2x} + e^{3x} + ...

Havil calls this a “non-legitimate binomial expansion” and it seems to me it can be generalised:

(1 - r^x)^{-1}= 1 + r^x + r^{2x} + r^{3x} + ...

as 1 = (1 - r^x)(1 + r^x +r^{2x}+r^{3x}+... )= 1 + r^x +r^{2x}+r^{3x}+...-r^x-r^{2x}-r^{3x}-...

And, indeed if we take x=-1, r=2 we get:

\frac{1}{1-2^{-1}} = 2 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} +... at the limit.

But if we have x \geq 0 it is divergent and the identity, which seems algebraically sound to me, breaks down. E.g., r=2, x=2:

\frac{1}{1-4} = -\frac{1}{3} = 1 + 4 + 8 + 16 + ...

So what is the flaw in my logic?

Was new maths really such a disaster?


English: Freeman Dyson
English: Freeman Dyson (Photo credit: Wikipedia)

On holiday now, so I fill my time – as you do – by reading books on maths.

One of these is Julian Havil’s Gamma: Exploring Euler’s Constant.

The book begins with a foreword by Freeman Dyson, in which he discusses the general failure, as he sees it (and I’d be inclined to agree), of mathematical education. He first dismisses learning by rote and then condemns “New Mathematics” as – despite its efforts to break free of the failures of learning by rote – an even greater disaster.

Now, I was taught a “new maths” curriculum up to the age of 16 and I wonder if it really was such a disaster. The one thing I can say is that it didn’t capture the beauty of maths in the way that the ‘A’ level (16 – 18) curriculum came close to doing. At times I really wondered what was it all about – at the age of 15 matrix maths seems excessively abstract.

But many years later I can see what all that was about and do think that my new maths education has given me quite a strong grounding in a lot of the fundamentals of computing and the applied maths that involves.

To the extent that this blog does have an audience I know that it is read by those with an interest in maths and computing and I would really welcome views on the strengths and weaknesses of the new maths approach.