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 April 18th, 2012, 11:20 PM #1 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory eulers constant Is it possible to prove that eulers constant is transcendental?
 April 18th, 2012, 11:31 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,184 Thanks: 481 Math Focus: Calculus/ODEs Re: eulers constant Yes. Do a search on "proof that e is transcendental" in your favorite search engine and you will find many proofs.
 April 18th, 2012, 11:42 PM #3 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: eulers constant No,not nepiers constant, i mean euler mascheroni constant
April 18th, 2012, 11:49 PM   #4
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Re: eulers constant

Ah, I wondered about whether you meant Napier's constant, which I've seen also referred to as Euler's constant. From Wikipedia:

Quote:
 The number ? has not been proved algebraic or transcendental. In fact, it is not even known whether ? is irrational. Continued fraction analysis reveals that if ? is rational, its denominator must be greater than $10^{242080}$

 April 19th, 2012, 12:06 AM #5 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: eulers constant Thanx. Can you show me how the limit of the difference of the harmonic series and natural logarithm output such a number????? I think your explanation will be much easier than other text books.
 April 19th, 2012, 12:18 AM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,184 Thanks: 481 Math Focus: Calculus/ODEs Re: eulers constant That is simply how it is defined, from my limited understanding of it. I'm sure someone who actually knows something about this constant will weigh in.
April 19th, 2012, 12:30 AM   #7
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Re: eulers constant

Quote:
I hope so.
Thank you for all that informations, its useful to me

April 19th, 2012, 05:18 AM   #8
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Re: eulers constant

Quote:
 Originally Posted by MarkFL That is simply how it is defined, from my limited understanding of it.
Yes, that's the usual definition.

I should mention that I think of $\gamma$ as the logarithm of a more fundamental constant, $e^\gamma.$ Somehow it's natural that the limit, involving logarithms as it does, has a constant which is the logarithm of the fundamental one.

I should mention at this point that $e^\gamma$ is also believed to be transcendental but not even known to be rational. If it is rational the denominator is large.

 April 19th, 2012, 07:52 AM #9 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: eulers constant But is it possible to derive euler mascheroni constant from that formula
April 19th, 2012, 01:21 PM   #10
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Re: eulers constant

Quote:
 Originally Posted by mathbalarka But is it possible to derive euler mascheroni constant from that formula
Yes, though it converges slowly.

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