View Full Version : integration of function

grav

2007-Jan-18, 06:29 AM

Does anyone know how to find the formula for the integration (limit) of the function,

[(ix)^2n]*(2n-1)!/(2n)! ?

The solution is real. It just alternates between negative and positive values. x is a variable. Spread out, it looks like this:

-x^2*(1/2) +x^4*(1/2)*(3/4) -x^6*(1/2)*(3/4)*(5/6) +x^8*(1/2)*(3/4)*(5/6)*(7/8) -x^10*(1/2)*(3/4)*(5/6)*(7/8)*(9/10) ...

For instance, the formula for the limit of 1/x+1/x^2+1/x^3+1/x^4 ... is L=x/(x-1)

grav

2007-Jan-18, 07:36 AM

Never mind. I got it. It's just L=1/sqrt(1+x^2)-1. Thanks anyway.

hhEb09'1

2007-Jan-20, 09:48 AM

Does anyone know how to find the formula for the integration (limit) of the function,

[(ix)^2n]*(2n-1)!/(2n)! ?You're not really integrating the function, right? You're finding the limit of the sum. But I think your formula should be different -- (2n-1)!/(2n)! is just 1/(2n). To match the expansion, I think you need [(-x^2)^n]*(2n-1)!/[2^(2n-1)*n!*(n-1)!]

Imaginary number is not necessary then, too.

grav

2007-Jan-20, 10:03 PM

You're not really integrating the function, right? You're finding the limit of the sum. But I think your formula should be different -- (2n-1)!/(2n)! is just 1/(2n). To match the expansion, I think you need [(-x^2)^n]*(2n-1)!/[2^(2n-1)*n!*(n-1)!]

Imaginary number is not necessary then, too.By integrating the function, I mean that I am finding the sum of all of the terms from n=1 to infinity, so yes, I'm finding the limit of the sum. Is there a difference with the way I'm saying it? You're right about the (2n-1)!/(2n)!=1/(2n). I'm not sure what the best way to write that should have been, but probably what you have. I'll have to look into it. I'm not sure if I found the limit of the sum from the expansion or the formula, either, so I'll have to check it again as well. Thank you tremendously.

Disinfo Agent

2007-Jan-20, 10:06 PM

By integrating the function, I mean that I am finding the sum of all of the terms from n=1 to infinity, so yes, I'm finding the limit of the sum. Is there a difference with the way I'm saying it?That's usually called "summing". :)

grav

2007-Jan-20, 10:12 PM

Yep. What you have checks out. Thanks again for that. :)

grav

2007-Jan-20, 10:15 PM

That's usually called "summing". :)With the roundabout way I do these mathematics sometimes, "slumming" would be more like it. :)

grav

2007-Jan-20, 11:06 PM

Okay, good. The limit is still the same as what I found, according to the expansion. Whew! I just wrote the formula for the expansion down wrong. Thanks again, hhEb09'1. You've shown me a couple of things here.

tusenfem

2007-Jan-21, 11:35 AM

Grav!

don't forget that there are books like "Abramowitch & Stegun" that are very very usefull in these matters.

hhEb09'1

2007-Jan-21, 12:55 PM

Grav!

don't forget that there are books like "Abramowitch & Stegun" that are very very usefull in these matters.Show him :)

What does it say about this particular sum?

tusenfem

2007-Jan-21, 02:58 PM

Show him :)

What does it say about this particular sum?

dunno. I have the book in my office and now I am sitting on the couch with a nice glass of Belgian beer.

hhEb09'1

2007-Jan-23, 05:51 AM

you have to put down the beer and go to work sometime :)

grav

2007-Jan-24, 04:02 AM

I think I have just found the best website there could possibly be for stuff like this. It is here (http://integrals.wolfram.com/index.jsp). It even gives explanations and graphs for some complex parts of the result, and it is the ultimate in user friendly. Cool. Me no have to think no more. :dance: (Actually, I could probably learn more experimenting with this than I would have taken the time to do on my own.)

publius

2007-Jan-24, 04:20 AM

I think I have just found the best website there could possibly be for stuff like this. It is here (http://integrals.wolfram.com/index.jsp). It even gives explanations and graphs for some complex parts of the result, and it is the ultimate in user friendly. Cool. Me no have to think no more. :dance: (Actually, I could probably learn more experimenting with this than I would have taken the time to do on my own.)

That the famous Mathematica program, which does symbolic math. I thought programming a computer to do *symbol manipulation* was amazing years ago, but now it has probably progressed to almost magic by comparison. I've got a couple of TI calculators that will do some symbolic math, but it's nothing like Mathematica.

IIRC, years ago (10+), Mathematica was very expensive. THere was cheaper one called MathCAD that I actually played around with, because the school could afford it. :) I have no idea what Mathematica costs now.

-Richard

hhEb09'1

2007-Jan-24, 06:47 PM

IIRC, years ago (10+), Mathematica was very expensive. The student edition, available in a lot of college bookstores for purchase by students, cost me around $100 twelve years ago, IIRC. I still use it sometimes.

I was always amazed to type in huge numbers into my PC and have them factored instantly.

publius

2007-Jan-24, 07:04 PM

Actually, that should be closer to 20 years ago than 10.

Time is flying.... This would've been around 1991 - 92. I remember the physics dept. had several PCs with MathCAD installed, but no Mathematica, and they said it was too expensive for their budget. Some depts. had it, IIRC.

The ins and outs of "student licensing" are something I don't understand. In some cases a student may be eligible for the academic discount, but the university itself might not....Of course, I'm always a student when I can be........

-Richard

hhEb09'1

2007-Jan-24, 07:14 PM

Actually, that should be closer to 20 years ago than 10.

Time is flying.... This would've been around 1991 - 92. It came bundled on all of our NeXT computers, early '92. Just less than 15 years ago--so, still closer to 10. :)

I had a 500 Mb removeable hard disk that I carried from computer to computer. I knew the location, and the availability times, of every NeXT on campus :)

Mathematica was released in '88.

publius

2007-Jan-25, 02:48 AM

You know, I'm getting as bad as my father and his bunch about time. When I was a kid back in the 70s, I'd hear them talk about the '50s and '60s as though it were yesterday. One of them even drove an old '55 Chevy pickup (his son still has it, but it doesn't run -- he says he's gonna restore it someday).

I thought that '55 pickup was absolutely *ancient*. How and why would anyone want to drive something that old....................

Now, I have an '86 Ford truck that I still drive. Now, in 2007 it's just as old as that '55 was back in the 70s! But it still seems different in my mind.

-Richard

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