The mathematical (and other) thoughts of a (now retired) math teacher,
Friday, 22 October 2010
A Nice Presentation of the Harmonic Series
Here is a nice video here about the Harmonic series. I would love for all my Pre-calc and calc students to see this video just to see the really nice proof by contradiction that the series is infinite. The paradox itself is just a bonus, and the fact that the squares add up to a value involving pi is a beautiful story that ought not to be popped out as an aside... sometimes we need to take the time to dig a little deeper into what Euler has given us... but a nice presentation all the same. Enjoy, but remember the adomonition that Borovik gave to his students,
" Today I said to the calculus students, "I know, you're looking at this series and you don't see what I'm warning you about. you look at it and you think, 'I trust this series. I would take candy from this series. I would get in a car with this series.' But I'm going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.
For more about the harmonic series, here and here and one more here, but surely there are more...use the search function.
Oh, was that quote by Borovik? I saw it at the Learning Curves blog, but I didn't realize she was quoting someone else. If you could give me the reference, I'll fix the quotation on my blog. Thanks!
Denise, I think I found that quote in a book by Julian Havel... perhaps "Impossible" or Non-plussed... If you can't find a link, I will look through my library to find them
My library loan system doesn't list a single book by Julian Havel. If you have time to look it up someday, I would appreciate it. I've subscribed to these comments, so if you don't get around to it for awhile, I'll get the notification whenever you do find the time.
Denise, That's a shame. Julian Havil is a master at Winchester College in England. A 30 year mathematics teacher, he brings university knowledge and classroom experience. His Publishers are Princeton University Press. I found the quote on page 117 of "Impossible" where he has the quote and then, "So wrote the Manchester University mathematician Alexandre Borovik of the series....." Hope that helps...
It was not only my local library that didn't have any books by Havel, but my library-loan system, which includes town, city, school, and community-college libraries from all over down-state Illinois. The system doesn't include the universities, and I don't think it stretches all the way to Chicago. But still, it's much more than just my little town.
I'll have to ask my librarian if she can find me the book outside the system...
Oh, was that quote by Borovik? I saw it at the Learning Curves blog, but I didn't realize she was quoting someone else. If you could give me the reference, I'll fix the quotation on my blog. Thanks!
ReplyDeleteDenise,
ReplyDeleteI think I found that quote in a book by Julian Havel... perhaps "Impossible" or Non-plussed... If you can't find a link, I will look through my library to find them
My library loan system doesn't list a single book by Julian Havel. If you have time to look it up someday, I would appreciate it. I've subscribed to these comments, so if you don't get around to it for awhile, I'll get the notification whenever you do find the time.
ReplyDeleteDenise,
ReplyDeleteThat's a shame. Julian Havil is a master at Winchester College in England. A 30 year mathematics teacher, he brings university knowledge and classroom experience. His Publishers are Princeton University Press.
I found the quote on page 117 of "Impossible" where he has the quote and then, "So wrote the Manchester University mathematician Alexandre Borovik of the series....."
Hope that helps...
Thank you!
ReplyDeleteIt was not only my local library that didn't have any books by Havel, but my library-loan system, which includes town, city, school, and community-college libraries from all over down-state Illinois. The system doesn't include the universities, and I don't think it stretches all the way to Chicago. But still, it's much more than just my little town.
I'll have to ask my librarian if she can find me the book outside the system...