Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

Evaluate the sum

Suppose that the real numbers

satisfying

Evaluate the sum

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Evaluate the sum

?

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Evaluate the sum

Hi tony123 and gAr;

This is a tough computational one. I am getting:

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Evaluate the sum

Hi bobbym,

Yes, that's it!
I made a mistake writing the answer.

How did you approach it?

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Evaluate the sum

Hi gAr;

With great difficulty and not much rigor. The way to answer it was by solving a 2002 x 2002 set of simultaneous equations. Only thing was M could not do it. A 4 million element matrix was too much memory. So what I did was I solved a 2 x 2, 4x4, ... 500 x 500. I played spot the pattern and then hoped that I could prove it by induction. No luck!

All I can say is that answer it is definitely correct for 2 x 2 all the way to 500 x 500. I am now trying to prove the answer with a Hilbert matrix which would allow me to just compute it.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Evaluate the sum

Hi bobbym,

Ok, All the best.

I too didn't prove it rigorously. I used cramer's rule to solve a 3x3 and 4x4 matrix and spotted a pattern.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."