Comments on: Why Convergence Matters http://leedsmathgeeks.com/2009/why-convergence-matters/ Math is exciting... math is fun! Sat, 21 Aug 2010 13:52:46 +0000 hourly 1 http://wordpress.org/?v=3.0 By: Vlad http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-1725 Vlad Sat, 21 Aug 2010 13:52:46 +0000 http://leedsmathgeeks.com/?p=47#comment-1725 @Chi: Exactly :) @Chi: Exactly :)

]]> By: Chi http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-1660 Chi Fri, 13 Aug 2010 01:26:59 +0000 http://leedsmathgeeks.com/?p=47#comment-1660 I know what you want to say. The sum of two divergent series may be divergent or may not. I know what you want to say.
The sum of two divergent series may be divergent or may not.

]]> By: Vlad http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-1655 Vlad Thu, 12 Aug 2010 10:00:47 +0000 http://leedsmathgeeks.com/?p=47#comment-1655 @Chi: You're absolutely right. 1+ 1 - 1 + 1 - 1 + 1... will never add up to 1/2. Neither will it add up to 1 or 0 <strong>because it's a divergent series and so doesn't have a sum</strong>. My last comment was only to point out the logical fallacy in your previous comment. Namely, the series... 1 + 1/2 - 1/3 + 1/4 - 1/5 + 1/6... converges. Yet taking all the odd terms, they form a divergent series, and so do all the even terms. That doesn't mean the full series itself is like adding two infinities :). Your conclusion may have been right, but your argument had a flaw, and I couldn't let you get away with that :D @Chi:

You’re absolutely right. 1+ 1 – 1 + 1 – 1 + 1… will never add up to 1/2. Neither will it add up to 1 or 0 because it’s a divergent series and so doesn’t have a sum.

My last comment was only to point out the logical fallacy in your previous comment. Namely, the series…

1 + 1/2 – 1/3 + 1/4 – 1/5 + 1/6…

converges. Yet taking all the odd terms, they form a divergent series, and so do all the even terms. That doesn’t mean the full series itself is like adding two infinities :) .

Your conclusion may have been right, but your argument had a flaw, and I couldn’t let you get away with that :D

]]> By: Chi http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-1654 Chi Thu, 12 Aug 2010 07:56:53 +0000 http://leedsmathgeeks.com/?p=47#comment-1654 @Vlad: Sum could be 0 and 1 but never be 1/2 2S=2-2+2-2+2... That will not equal to 1 Try to sketch S = 1 – 1 + 1 – 1 + 1 – 1 + … It will show you that does not converge If you do this S = 1 – 1 + 1 – 1 + 1 – 1 + … S = 1 – (1 – 1 + 1 – 1 + 1 – 1 + …) it show that you may think first S is equal to 1 the second S is equal to 0 or the first is 0 the second is 1 you are a really geek thanks! @Vlad:
Sum could be 0 and 1 but never be 1/2
2S=2-2+2-2+2…
That will not equal to 1
Try to sketch S = 1 – 1 + 1 – 1 + 1 – 1 + …
It will show you that does not converge

If you do this
S = 1 – 1 + 1 – 1 + 1 – 1 + …
S = 1 – (1 – 1 + 1 – 1 + 1 – 1 + …)
it show that you may think first S is equal to 1
the second S is equal to 0
or the first is 0
the second is 1

you are a really geek
thanks!

]]> By: Vlad http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-1642 Vlad Tue, 10 Aug 2010 14:24:41 +0000 http://leedsmathgeeks.com/?p=47#comment-1642 @Chi: I have a series for you to think about: 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 + ... Now, clearly, this series converges. But what happens if you take only the positive terms, and only the negative terms? (If you can't figure it out, try googling "harmonic series") @Chi:

I have a series for you to think about:

1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + …

Now, clearly, this series converges. But what happens if you take only the positive terms, and only the negative terms?

(If you can’t figure it out, try googling “harmonic series”)

]]> By: Chi Zhang http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-1635 Chi Zhang Tue, 10 Aug 2010 05:00:05 +0000 http://leedsmathgeeks.com/?p=47#comment-1635 S=1-1+1-1+ S=(1+1+1+1...)+(-1-1-1...) you can not add pos infinity and a neg infinity together S=1-1+1-1+
S=(1+1+1+1…)+(-1-1-1…)
you can not add pos infinity and a neg infinity together

]]> By: Amy http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-1308 Amy Thu, 03 Jun 2010 23:51:23 +0000 http://leedsmathgeeks.com/?p=47#comment-1308 To say divergent series are almost useless is wrong - they have great use in applied mathematics as asymptotic expansions, or to provide numerical approximations via Borel summation etc. To say divergent series are almost useless is wrong – they have great use in applied mathematics as asymptotic expansions, or to provide numerical approximations via Borel summation etc.

]]> By: Why is 0.99999… = 1? | Leeds Math Geeks http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-31 Why is 0.99999… = 1? | Leeds Math Geeks Tue, 17 Mar 2009 11:48:15 +0000 http://leedsmathgeeks.com/?p=47#comment-31 [...] course, whenever you hear a fancy proof like this, you have to be very careful. Like I showed in Why convergence matters, you can sometimes use seemingly impeccable logic to show complete [...] [...] course, whenever you hear a fancy proof like this, you have to be very careful. Like I showed in Why convergence matters, you can sometimes use seemingly impeccable logic to show complete [...]

]]> By: richieacc http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-17 richieacc Tue, 24 Feb 2009 17:32:53 +0000 http://leedsmathgeeks.com/?p=47#comment-17 Thanks Vlad, makes sense now. Can't explain why it made no sense earlier :) Thanks Vlad, makes sense now.

Can’t explain why it made no sense earlier :)

]]> By: Dr Morrison http://leedsmathgeeks.com/2009/why-convergence-matters/comment-page-1/#comment-16 Dr Morrison Tue, 24 Feb 2009 15:16:15 +0000 http://leedsmathgeeks.com/?p=47#comment-16 To say divergent series are almost useless is wrong - they have great use in applied mathematics as asymptotic expansions, or to provide numerical approximations via Borel summation etc. To say divergent series are almost useless is wrong – they have great use in applied mathematics as asymptotic expansions, or to provide numerical approximations via Borel summation etc.

]]>