I want to show some school students about the sum of the harmonic series diverging and I need some nice interpretation for this. I'll preferably love to have a figure.
Thanks in advance!
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3Isn't the standard proof enough (grouping elements that add up to 1 or more)? http://math.stackexchange.com/questions/172504/why-do-we-say-the-harmonic-series-is-divergent – Lepidopterist Mar 16 '13 at 18:36
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@Lepidopterist: I do not seem a proof exactly, but more of a figure where you can see the obvious! – Guru Mar 16 '13 at 18:37
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Guru, then make a really good diagram up to $n=8$ – Will Jagy Mar 16 '13 at 18:39
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2http://prairiestate.edu/skifowit/harmapa.pdf – Lepidopterist Mar 16 '13 at 18:39
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8I doubt you'll find a "non-mathematical" way, given that this is, after all, maths. – L. F. Mar 16 '13 at 18:44
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2Following @Lepidopterist, here are 22 more proofs. – Antonio Vargas Mar 16 '13 at 18:50
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Look at the (first) book Proofs without words. – André Nicolas Mar 16 '13 at 18:51
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I'll need to elaborate on one of the "visual proofs" in the second pdf, I think. I was looking for one of those in the answer section actually. – Guru Mar 16 '13 at 18:51
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How much math have these students previously had? – liamdalton Mar 16 '13 at 19:27
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One way is to bound the series below by a series that more immediately looks infinite. We round down $1/n$ to largest fraction of the form $1/2^k,k\in\mathbb N$. This is illustrated as such: \begin{align*} &1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\ldots\\ \geq&1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\ldots \end{align*} Hopefully that presentation is clear enough. We may combine terms in this lower sum to see (loosely) that we are adding up $1/2$ an infinite number of times. This shows that the sum diverges, at least sufficiently for illustrative purposes.
Ian Coley
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