Happy Birthday, Leonhard Euler! - Page 3

mfgoode · 2007-05-06, 16:58

Quote:

Originally Posted by jinydu

Sure it can. For instance, take $\frac{\sqrt{2}}{\sqrt{2}}$ .

Actually, $\frac{\zeta(26)}{\pi^2}$ is a rational multiple of $\pi^{24}$ and is hence irrational, while $\zeta(3)$ itself is irrational.

After computing the first 10 terms of the continued fraction expansion for $\frac{\zeta(26)}{\pi^2}$ and combining it into a simple fraction, I get:

$\frac{158157}{1560947}$

Quote:

Originally Posted by jinydu

Sure it can. For instance, take $\frac{\sqrt{2}}{\sqrt{2}}$ .

Can you give another example not coming to 1 but any other integer?

I'm extremely sorry jinydu. That was a typographical error in my post.

I meant [tex]\frac{\zeta(26}{\pi^26}[\tex] i.e. Zeta 26/pi^26 in case the tex doesnt show up

Now what is the fraction?

Mally

P.S. BTW what went wrong with my tex language?

jinydu · 2007-05-06, 18:07

Quote:

Originally Posted by mfgoode

Can you give another example not coming to 1 but any other integer?

How about $\frac{n\sqrt{2}}{\sqrt{2}}$ for any integer $n$ .

Quote:

Originally Posted by mfgoode

I meant [tex]\frac{\zeta(26}{\pi^26}[\tex] i.e. Zeta 26/pi^26 in case the tex doesnt show up

Now what is the fraction?

$\frac{1315862}{11094481976030578125}$

Quote:

Originally Posted by mfgoode

P.S. BTW what went wrong with my tex language?

Use /tex not \tex

mfgoode · 2007-05-08, 12:34

Quote:

Originally Posted by jinydu

How about $\frac{n\sqrt{2}}{\sqrt{2}}$ for any integer $n$ .

$\frac{1315862}{11094481976030578125}$

Use /tex not \tex

Your fraction is correct.
Thanks Jinydu.

Mally

fivemack · 2007-05-09, 11:57

There's a useful trick for special-values-of-functions:

go to http://integrals.wolfram.com/index.jsp

and ask to integrate Zeta[26]

and the output comes out as 1315862 pi^26 x / 11094481976030578125

I don't think this is the intended use of integrals.wolfram.com, but I don't know a full-fat Web Mathematica.

mfgoode · 2007-05-09, 16:10

Quote:

Originally Posted by fivemack

There's a useful trick for special-values-of-functions:

go to http://integrals.wolfram.com/index.jsp

and ask to integrate Zeta[26]

and the output comes out as 1315862 pi^26 x / 11094481976030578125

I don't think this is the intended use of integrals.wolfram.com, but I don't know a full-fat Web Mathematica.

Thank you fivemack for the 'useful trick' for summing up of series. I have noted it down

At first glance I wondered how can one integrate a series without a variable, so I tried the URL you gave disbelievingly. Seeing an 'x' in the answer I caught on as strictly speaking it is actually the summing up of Zeta[26]x^0 hence on integration one must get x. This is a fine example for summation of series.

Well we all live and learn which is an unending process!.

The next logical step is to plug in Zeta (3) and lets see what comes up.
I haven't done it right now and will let you know what comes up.

Great Guns, fivemack!

Mally

P.S. I got xZeta[3] Not even a decimal answer which jinydu got from his site.

jinydu · 2007-05-10, 00:56

Quote:

Originally Posted by fivemack

There's a useful trick for special-values-of-functions:

go to http://integrals.wolfram.com/index.jsp

and ask to integrate Zeta[26]

and the output comes out as 1315862 pi^26 x / 11094481976030578125

I don't think this is the intended use of integrals.wolfram.com, but I don't know a full-fat Web Mathematica.

That shouldn't be too much of a surprise. The integral of a constant is the variable times that constant.

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2007-05-09, 11:57	#26
fivemack (loop (#_fork)) Feb 2006 Cambridge, England 2³×11×73 Posts	There's a useful trick for special-values-of-functions: go to http://integrals.wolfram.com/index.jsp and ask to integrate Zeta[26] and the output comes out as 1315862 pi^26 x / 11094481976030578125 I don't think this is the intended use of integrals.wolfram.com, but I don't know a full-fat Web Mathematica.