Help?

[while1]

Evaluate:

∞
∑ ( n/(n+1)! )
n=1


Now, I know that the damn thing is equal to 1, however I'm having trouble showing/proving this... I'd greatly appreciate any help.  Thanks.

[rabbit]

1/2 + 1/6 + 1/24 ... + 0 ~= 1
It's approximately 1, IIRC.

[edit]
Take the limit.

[Sidoh]

It's a telescoping series.  It's sort of tricky to get it in that form.

[Ender]

1/2 + 1/6 + 1/24 ... + 0 ~= 1
It's approximately 1, IIRC.

[edit]
Take the limit.

You've misrepresented the series. The numerator is a variable. And "take the limit" isn't really helpful.

Prove that the series of n / (n + 1)! from n = 1 to infinity converges to 1.

Proof:

We are going to prove that the series converges to 1 by mathematical induction. Let S(n) be the nth partial sum of the series.

Assume that S(n) = [(n+1)! - 1] / (n + 1)!. Then S(n+1) = [(n+1)! - 1] / (n + 1)! + (n+1) / (n+2)!. Multiply the lefthand term by (n + 2) / (n + 2) to get a common denominator. Simplifying, you get S(n+1) = [(n+2)! - 1] / (n+2)!. Therefore, if S(n) is true for some n, then it is true for n + 1. 

We now consider the anchor case, n = 1. S(1) = [(1 + 1)! - 1] / 2 = 1/2.

Therefore, S(n) is true for all n, as proved by induction.

Now consider the limit of S(n) as n goes to infinity. lim S(n) as n->infinity = 1. Therefore the partial sum converges to 1.

[Ender]

Clever solution, Sidoh. Just to make your solution more readily visible:

You can write out the 2nd partial sum in that format, getting.

(2/2 - 1/2) + (3/6 - 1/6)

You see that the middle terms cancel out, and that you end up getting 2/2 - 1/6. This will be true for any nth partial sum, and you will end up getting the first term minus the last term, that is 2/2 - 0 = 1, since the right-hand term will go to 0 as seen by the limit.

[while1]

I was totally approaching the problem all the wrong ways. i.e. I tried representing the factorial as an integral, which ended up being Gamma functions >_<;

Thanks Sido and Ender.

@rabbit misread the series.

@Sido  That's what I forgot.  I tried to convert it to a geometric and harmonic series but forgot to try telescoping >_<;  Nifty solution, thanks. 

@Ender different method, same conclusion, thanks!

EDIT:  Sido do you use a program to create those images?  If so, may I ask what it is?

[Ergot]

EDIT:  Sido do you use a program to create those images?  If so, may I ask what it is?

Latex.

[Sidoh]

I was totally approaching the problem all the wrong ways. i.e. I tried representing the factorial as an integral, which ended up being Gamma functions >_<;

Thanks Sido and Ender.

Sido's is quite a nifty solution.

Anyways, thanks.

EDIT:  Sido do you use a program to create those images?  If so, may I ask what it is?

No problem.

As Ergot mentioned, the images are generated through a Latex script I have on my server.  It renders the input and then chains it through an otherwise painful series of conversions (input(which is usually a simplified form of latex)->tex->dvi->gif, I think). 

Anyway, Latex is pretty easy to use.  You can access the script here.  There's links to Latex documentation on that page as well.  The images are generated on the fly (if you take a look at the paths to the ones I posted, you'll probably be able to see that), so you can either save them or post the URL straight into a post (with BBCode, obviously).

Oh, here's a post I made a while ago with some examples.

[while1]

Nice!  I'll definitely be using the script whenever I boot into Windows... actually methinks I'll just use the script while in linux too.

[Sidoh]

Nice!  I'll definitely be using the script whenever I boot into Windows... actually methinks I'll just use the script while in linux too.

Well (if you didn't get the idea from poking around with it), it's totally web based.  As long as you have a browser that supports some minimal javascript, then you can use it under any OS.

By the way, welcome to x86.  I forgot to mention that before.  If you plan on sticking around, make a post in the introduction forum if you'd like. :)

[while1]

Nice!  I'll definitely be using the script whenever I boot into Windows... actually methinks I'll just use the script while in linux too.

Well (if you didn't get the idea from poking around with it), it's totally web based.  As long as you have a browser that supports some minimal javascript, then you can use it under any OS.

By the way, welcome to x86.  I forgot to mention that before.  If you plan on sticking around, make a post in the introduction forum if you'd like. :)

I was more so referring to the fact that I was going to download and install the LaTeX package in linux and write a program/script to do what your script does locally, and just use your script when in Windows since at a quick glance it looks like it's more of a hassle to download and install the win32 port of it.  However why reinvent the wheel for a task that's relatively low priority to me, right?

And thanks for the welcome.  I've had this account created for awhile, I just never really used it... I believe I was referred here by Ender, however before that I've seen references to it from time to time on the vL forums.  I guess I'll make an introduction thread in intro forum soon.

[Sidoh]

I was more so referring to the fact that I was going to download and install the LaTeX package in linux and write a program/script to do what your script does locally, and just use your script when in Windows since at a quick glance it looks like it's more of a hassle to download and install the win32 port of it.  However why reinvent the wheel, right?

Ah, I see.  Almost all Linux distributions come with Latex preinstalled.  When I've used it on Linux, I just use vim to write the tex, run latex on the file (which produced a dvi).  There's a converter (again, I'm pretty sure most distributions of Linux come with it) called dvi2ps, which converts that to a postscript document.

As you suggested, in Windows, composing latex documents is a much more painful process (or at least installing the latex processor).  I found some guide with step-by-step instructions on how to do it, but I'm not completely sure that it was the least convoluted way to achieve latex on windows.  The computers in the mathematics department at my university (which are windows machines) have some other editing environment installed (I use xemacs on windows), but I seem to recall the same port of latex appearing in the programs list as well (miktex).

In any case, a web script seems like the most useful form of this sort of thing since its rarely useful to be able to create the images if you don't want to post them on the web (if you're using latex to write a paper or something, you usually want to write the whole paper in latex, not just embed images into some other kind of document).

[rabbit]

What the hell did I see when I posted?  Hmm...  I must be going senile.

[Sidoh]

What the hell did I see when I posted?  Hmm...  I must be going senile.

lol.  Misreading the problem sucks. :(

Oh, and the series you were getting at converges to , according to Maple.

[while1]

Yeah, that makes sense since