Math Is Fun Forum

RauLiTo

Member

From: Bahrain

Registered: 2006-01-11

Posts: 142

infinite series

find
the value of this infinite series

Last edited by RauLiTo (2007-06-21 10:53:12)

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ImPo$$!BLe = NoTH!nG
Go DowN DeeP iNTo aNyTHinG U WiLL FinD MaTHeMaTiCs ...

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: infinite series

(i think the above is right)

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George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: infinite series

However this formula almost only makes it harder to solve.
I prefer the original one.:/

Last edited by George,Y (2007-06-21 13:09:59)

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X'(y-Xβ)=0

shocamefromebay

Member

Registered: 2007-05-30

Posts: 103

Re: infinite series

hmmm
what are the possiblities that this does not get infinitly close to one number
and just constantly increases?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: infinite series

Whoops, forgot to come back to this problem.

what are the possiblities that this does not get infinitly close to one number
and just constantly increases?

None.  It can be easily shown that each term is less than 1/n^2, and the sum of 1/n^2 converges.  Thus, this must converge.

So we may say:

The rest is just algebra.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: infinite series

No, it definitely converges to something. The series is equivalent to this:

Each term in that sum is smaller than the equivalent term in

, but that sum is finite.

Edit: Heh, post collision and we say almost exactly the same thing. Weird.

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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: infinite series

Edit: Heh, post collision and we say almost exactly the same thing. Weird.

I wouldn't say weird.  1/n^2 is a very popular series and the original series is 1 / something*something else*something else.  It only makes sense to compare this to 1/n^2.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

shocamefromebay

Member

Registered: 2007-05-30

Posts: 103

Re: infinite series

well
i made a prog on my calc about 5 mins ago to do sequences for me
and usually if the number converges
it jsut rounds and give me that number that it converges to
and i put this one into it
and it just kept increasing
for like 45 mins
then i just turned it off
and the calc couldnt even turn the number into fractions form
so i think if its byond the power of my calcy
its beyond the power of me
but this shows that this problem
is pretty difficult

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: infinite series

shocamefromebay, no trust me, it DOES converge, you program is wrong:

simple AS2 script:

var s:Number = 0;
var i:Number = 1;
function onEnterFrame():Void
{
   for(var j:Number = 0; j<5000; j++)
   {
       s+= 1/(3*i*(3*i-1)*(3*i-2));
       i++;
   }
    trace(i+" : "+s);
}

printing a value every 5000 times, you get to:
140001 : 0.178796768890753

from which point it doesn't change

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: infinite series

Probably too late to say this as we now have an answer, but it simplifies a bit to give:

I'm not sure if that helps any, but it would be nice to find an exact answer if we can.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: infinite series

Anyone know why what mathsyperson did is wrong?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: infinite series

The individual part sums are all divergent?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: infinite series

Well, that's part of it.

Mathsyperson rewrote it as an alternating series, which is fine.  But an alternating series can only have its terms reordered if it is absolutely convergent.  This one isn't.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

shocamefromebay

Member

Registered: 2007-05-30

Posts: 103

Re: infinite series

shocamefromebay, no trust me, it DOES converge, you program is wrong:

this doesnt necessarily mean that its wrong
it just means that it has reached the point where it converged yet

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: infinite series

it quickly converges in the small number of decimal places, after that it takes some time for the rest of the decimal places to follow, but even after say 10 iterations, its clear its converging;

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The Beginning Of All Things To End.
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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: infinite series

Funny luca, I would probably say the same thing about 1/n.  Your eyes may deceive you.  Don't trust them.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: infinite series

but 1/n doesn't do that, it constantly changes, maybe a few decimal places remain consistent, but then they change a few iterations later, well, maybe i exxagerated a bit on 10 iterations , but the first 100 iterations of the series printing every 5 iterations: after 5 iterations the first 3 decimal places never change, after 15 iterations it has the fourth decimal place, then it starts taking longer and longer to converge.

5 0.1781080031080031
10 0.17861794090836494
15 0.1787163312115937
20 0.17875125640305384
25 0.1787675393766264
30 0.17877642383782857
35 0.17878179703796668
40 0.17878529205149957
45 0.17878769217497648
50 0.17878941118940814
55 0.1787906843899262
60 0.17879165359266566
65 0.1787924084010432
70 0.17879300768311585
75 0.17879349140653122
80 0.17879388747893882
85 0.17879421586516644
90 0.17879449115322782
95 0.17879472420280887

after 140,000 iterations: 0.17879676889075347
after 1,000,000 iterations: 0.17879676889075347
after 100,000,000 iterations: 0.17879676889075347

pretty safe to say it converges.

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actually, why doesn't 1/n converge? because 1/n gets smaller and smaller towards 0, i would have thought it would converge?

Last edited by luca-deltodesco (2007-06-25 17:56:56)

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The Beginning Of All Things To End.
The End Of All Things To Come.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: infinite series

actually, why doesn't 1/n converge? because 1/n gets smaller and smaller towards 0, i would have thought it would converge?

That's a good question and I wish I had a sufficient answer.  Basically, it just doesn't converge fast enough.  But I still find series like this mysterious.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

shocamefromebay

Member

Registered: 2007-05-30

Posts: 103

Re: infinite series

teh solution for this one is e
but before theinvention/discovery/creation of e
no one could have figured it out
what are the possibilities taht this infinite series is like the one shown above
that there really is no rational answer taht we can put in fractional form?