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 Post subject: Calculus (13)Posted: Sat, 21 Jun 2008 21:36:59 UTC
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Evaluate:
.......................................................................................................
Remark: The limit is a special case of a general result. Have fun! :)

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 Post subject: Posted: Tue, 16 Sep 2008 05:30:54 UTC
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Joined: Thu, 11 Sep 2008 01:59:14 UTC
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I know the result of that integral and how to calculate the limit but... how do I put all those symbols that you used

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 Post subject: Posted: Tue, 16 Sep 2008 05:40:11 UTC
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illusion: if you click the quote button above commutative's post, then you can see how he wrote all those symbols.

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 Post subject: Posted: Tue, 16 Sep 2008 23:02:32 UTC
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I also recommend this link illusion.

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 Post subject: Posted: Mon, 22 Sep 2008 16:11:33 UTC
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Haven't solved it, but am I right in thinking it can be 'simplified' to:

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 Post subject: Re: Calculus (13)Posted: Tue, 23 Sep 2008 00:43:37 UTC
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I think I have it
Spoiler:

Make the substitution so it is transformed into
The integral converges to 1 as n approaches infinity, since
So now we have
The sum is
Dividing this by nn and taking the limit as n approaches infinity gives the answer as 1/2.

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 Post subject: Re: Calculus (13)Posted: Tue, 23 Sep 2008 00:53:41 UTC
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simplethinker wrote:
I think I have it
...
The integral converges to 1 as n approaches infinity, since

Doesn't depend on ?

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 Post subject: Re: Calculus (13)Posted: Tue, 23 Sep 2008 01:27:59 UTC
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Joined: Fri, 9 Nov 2007 00:04:38 UTC
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Zone Ranger wrote:
simplethinker wrote:
I think I have it
...
The integral converges to 1 as n approaches infinity, since

Doesn't depend on ?

k itself doesn't depend on n, only the limit of the sum. I split the equation into two parts with the basic property of limits that if exist then

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"The reason that every major university maintains a department of mathematics is that it is cheaper to do this than to institutionalize all those people."

Some people live their lives by the KISS principle (Keep It Simple Stupid), but I prefer a slightly different one:
Make It Fun And Complicate It Further, or MIFACIF. Ever try and find the volume of a cube in spherical coordinates?

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 Post subject: Re: Calculus (13)Posted: Tue, 23 Sep 2008 02:10:24 UTC
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simplethinker wrote:
Zone Ranger wrote:
simplethinker wrote:
I think I have it
...
The integral converges to 1 as n approaches infinity, since

Doesn't depend on ?

k itself doesn't depend on n, only the limit of the sum. I split the equation into two parts with the basic property of limits that if exist then

How did you get the limit next to the integral?

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 Post subject: Re: Calculus (13)Posted: Tue, 23 Sep 2008 03:22:25 UTC
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I'm not quite sure what you mean by "the limit next to the integral". If you mean how did I get that particular limit as part of the equation:

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Some people live their lives by the KISS principle (Keep It Simple Stupid), but I prefer a slightly different one:
Make It Fun And Complicate It Further, or MIFACIF. Ever try and find the volume of a cube in spherical coordinates?

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 Post subject: Re: Calculus (13)Posted: Tue, 23 Sep 2008 03:24:14 UTC
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simplethinker wrote:
I'm not quite sure what you mean by "the limit next to the integral". If you mean how did I get that particular limit as part of the equation:

you can't do that...it doesn't work that way..also the answer isn't 1/2

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 Post subject: Posted: Tue, 23 Sep 2008 03:37:52 UTC
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It doesn't work that way? It says it right here (on the SOS math site) as well as mathworld, wikipedia and one of my old calculus books. Since both limits exist it's valid. Granted my answer might not be correct, but I don't think it's that step where I messed up.

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"The reason that every major university maintains a department of mathematics is that it is cheaper to do this than to institutionalize all those people."

Some people live their lives by the KISS principle (Keep It Simple Stupid), but I prefer a slightly different one:
Make It Fun And Complicate It Further, or MIFACIF. Ever try and find the volume of a cube in spherical coordinates?

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 Post subject: Posted: Tue, 23 Sep 2008 03:40:38 UTC
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simplethinker wrote:
It doesn't work that way? It says it right here (on the SOS math site) as well as mathworld, wikipedia and one of my old calculus books. Since both limits exist it's valid. Granted my answer might not be correct, but I don't think it's that step where I messed up.

simplethinker wrote:
...

you cant pass the limit through this sum...that is the rule you are breaking.

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 Post subject: Posted: Tue, 23 Sep 2008 03:54:38 UTC
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Ah, that makes sense. I knew it was too easy
Thanks for catching it. I was playing around with the different parts and didn't pay attention to the rest of the equation. Now that I think about it, since the integral depends on 'k' I couldn't have split it up into a product.
I still have this feeling that taking the limit makes the integral equal one.

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"The reason that every major university maintains a department of mathematics is that it is cheaper to do this than to institutionalize all those people."

Some people live their lives by the KISS principle (Keep It Simple Stupid), but I prefer a slightly different one:
Make It Fun And Complicate It Further, or MIFACIF. Ever try and find the volume of a cube in spherical coordinates?

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 Post subject: Posted: Tue, 23 Sep 2008 03:56:57 UTC
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simplethinker wrote:
Ah, that makes sense. I knew it was too easy
Thanks for catching it. I was playing around with the different parts and didn't pay attention to the rest of the equation. Now that I think about it, since the integral depends on 'k' I couldn't have split it up into a product.
I still have this feeling that taking the limit makes the integral equal one.

the answer is close to 1/2...

try finding upper and lower bounds for the integral part

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