Official TSR Mathematical Society

Find $\displaystyle\sum^{\infty}_{n=0} (\frac{1}{n!})^2 + 2\sum^{\infty}_{n=0}(\sum^{\infty}_{m=n+1} \frac{1}{n!m!})$

edited

Apologies for this is off topic...

How to you embed maths characters into a TSR post? Do you do it on an external website and C+P?

I tried to search for a similar question, to no avail.

Okay, time for a full solution. This is the nastiest piece of work I've ever attempted to solve I think Hope I'm right... I'll be very interested in seeing your method Dystopia, because I think there will be a variety of methods that will lead to the answer - and I don't think I've taken the shortest (and I've probably spent 3-4hrs on the integral!)

[snip]

Erhm, now I should sub back for my t^2=tan(x/2) substitution, but I'll leave that for now I think!

Yes, great fun (At least I managed to avoid physics revision for a couple of hours!)

My substitution was efficient in getting it down to a rational function, but unfortunately it was a particularily nasty rational function (at some stage even mathematica couldn't do it, as I was cheking if I'd done it right).

Your method is obviously a lot more efficient, and I never spot the kpi+x substitutions sometimes required in STEP, and intergals like this - very nice approach!

edit: Incidentally, if you want a simple problem to play with, see if you can find a recurrence with $\displaystyle\mathcal{L} \left\{f^n(t)\right\}$ where f^n is the nth derivative.(And then you can solve it explicitly, and prove by induction if you feel like it) I stumbled upon that one yesterday, but to my disappointment I found it on wikipedia

I don't know much about Laplace transformations either; you only need the definition of a laplace transform i.e. $\displaystyle\mathcal{L}\left\{f(t)\right\}= \displaystyle\int_0^{\infty} e^{-st}f(t)dt$ and then you can (quite easily) show what is in my spoiler. Then, play around with how that spoiler generalises (start by looking at L{f''(t)}=? to get an idea of what is happening).

I only found this as an exercise in my book (the part in the spoiler and the L{f''(t)} case) - it's not that interesting.

Oh, and I think that nice result you have there is very closely related to the one I am after, both seem to rely on repeated integration by parts (except on the other integrand)...

Is this what you were looking for:

$\displaystyle \mathcal{L}\left\{f^{n} (t)\right\} = s^{n} \mathcal{L}\left\{f(t)\right\} - \sum_{r=1}^{n} s^{n-r} f^{r-1}(0)$

If so, you're right - setting s = 1, and letting n tend to infinity gives the same result (again, assuming various things tend to zero).

By the way, do you know anything about double integrals? When I tried to find a function such that $\displaystyle \int^{\infty}_{0} e^{-x} f(x) \; \mathrm{d}x = 1 - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}$, I think I ended up solving a differential equation to get $\displaystyle f(x) = \frac{1}{2\sqrt{x}} \int^{x}_{0} \frac{e^{-t}}{\sqrt{t}} \; \mathrm{d}t$. I don't know anything about double integrals at all, but I assume that this is related to the result $\displaystyle \int^{\infty}_{0} e^{-x^{2}} \; \mathrm{d}x = \frac{\sqrt{\pi}}{2}$?

Oh, and when I was avoiding doing physics, I noticed that considering the real and imaginary parts of $\ln(1 + e^{ix})$ gives you a couple of nice identities. One of them provides yet another incredibly dodgy way of finding the sum of the reciprocals of the squares, whereas the other provides an interesting way of finding the classic integral $\displaystyle \int^{\frac{\pi}{2}}_{0} \ln(\cos x) \; \mathrm{d}x$