Explaining how 1 + 2 + 3 + . . . can possibly equal -1/12 to a kid

When I did the my biographies for my kids last week my older son said that the thing in math that he’s see but that he does not believe is this equality:

1 + 2 + 3 + 4 + . . . . = -1/12

This sum was made popular by a Numberphile video a couple of years ago (which now has over 4 million views!):

 

there have also been several good follow ups. For example this video with Ed Frenkel which was also produced by Numberphile:

 

and this video by Mathologer which is absolutely excellent:

 

I spent some time today trying to think about how to discuss this series with my older son. I’m glad that he is bothered by the result – it is obviously very very strange. Obviously I can’t go into the details about the Riemann Zeta function with him, but I still think there’s some what to help him make some sense of the series. So, I spent the day reviewing some ideas in G. H. Hardy’s book “Divergent Series.” Here are a few passages that caught my eye:

(a) Book Cover

I don’t remember where I heard about this book. My best guess is that it was mentioned in Jordan Ellenberg’s “How Not to be Wrong” in the section about Grandi’s series. Unfortunately I only have the audiobook version of “How not to be Wrong” and don’t know how to search it!

(b) first passage

The remark beginning at “It is plain . . . ” caught my attention.  This is right at the beginning of the book – section 1.3.   The statement:

“it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition.”

also felt very powerful to me.

(c) second passage

The continuation of the previous page is also important – the point about Cauchy was definitely mentioned in “How not to be Wrong” as well.

(d) third passage

 

For the third passage we have to go much later in the book – nearly to the end, in fact.  The passage here – 13.10.11, in particular – shows the strange result.  Not in a Numberphile video, or some other internet video, but in a math textbook by G. H. Hardy:

(e) fourth passage

Finally – and this really is just about the last page of the book – section 13.17 provides a word of caution and an example of what can go wrong playing around with these divergent infinite series.

 

So, I’m going to spend the next few days and maybe even the next few weeks thinking about how to share some sort of idea about this strange series with my son.  I’ll welcome any suggestions!

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