/sci/ - Science & Math

>>7149585
Well it has a finite number of terms for starters.

>finite series
>convergent

welp I meant for very large N

There any good resources on convergent/divergent stuff? My prof doesn't teach very well (44% class average going into the final), and was wondering if there's any other sites that can save me.

>>7149605
this.

Btw. you can write it as <span class="math">\sum_{n=1}^n\frac{k^3}{k!}[/spoiler], which is arguably nicer.
It's also bounded by what pops up in combinatorics:

http://en.wikipedia.org/wiki/Dobinski%27s_formula

>>7149611
>very large N

<div class="math">k^2 = (k-1)(k-2) + 3(k-1) + 1 </div>
<div class="math">\sum_{k=1}^\infty \frac{k^2}{(k-1)!} =\sum_{k=3}^\infty \frac{1}{(k-3)!} + \sum_{k=2}^\infty \frac{3}{(k-2)!} +\sum_{k=1}^\infty \frac{1}{(k-1)!}=5 \sum_{k=0}^\infty \frac{1}{k!} = 5e</div>

>>7149585
If lim n-> inf use the ratio or root test.

>>7149617
sum over k, I mean.

>>7149629
cool.

>>7149585
OP, did this problem pop up when dealing with Poisson distributions, by any chance?

>>7149585
for k>a, a being some specific number
k^2/(k-1)! < 1/k
by comparison test, it converges

>>7149629
Thanks.
>>7149638
It came up when I tried to solve a recurrence relation.

>>7149646
This is wrong.

>>7149629

Huh, that's pretty nifty.

So, quick question, liquor has gotten to me, what is the complex conjugate for this: psi = (exp-Zr/2anot)(cos theta) I have the variables r and theta. It is fucking with me and I need it for an expectation value. Are there threads for working out shit like this? Can't remember how to do it for two variables.

>>7150369
type it into wolfram...

>>7149629
very intuitive
reminds me of pic related

>>7150369
also im guessing you are in quantum 1, but its puzzling you ask this question so late in the semester.. you know a complex conjugate just swaps the sign of all i's right?.... your wavefunction has no i's... its just psi^2. plz go to class. Also its going to bite you in the ass you didnt pay much attention in linear algebra and google euler's formula plz

>>7150427
actually it is, you just make the ansatz
<div class="math">(k-1)(k-2) = k^2 - 3k + 2 </div>
because you want to express the <span class="math">k^2[/spoiler] in terms of descending numbers to cancel with the factorial and then continue with the remainder <span class="math"> 3k - 2 [/spoiler] in the same way

>>7150427
What book is that page from?

>>7150427
This is why I dislike maths, shit like that seems like you have to know the answer before finding it. Fuck this wizardry.

>>7149749
please explain, i want to learn

>>7149613
just need a bit of intiution, so that you can look at a series like that an be 99% sure you are showing convergence rather than divergence.

My other tip is there is a general heirachy of convergence speed when it comes to simple functions. n^l<n^k<x^n<n!<n^n for all x>1, l>k, n==>inf

Sorry for shitty formatting.

>>7149629
>ohshitnigga.jpg

>>7149585
that depends; what is the value of n?

>>7150690
This.
>ok, time to simplify <span class="math">k^2/(k-1)![/math[
>can't cancel shit because of factorial
>write denominator as (k-1)(k-2)...1
>still can't cancel shit, numerator is wrong
>recall calc 1 (integrating rational functions) or intro to probability (calculating factorial moments)
>realize you can just rewrite <span class="math">k^2[/spoiler] to make the things cancel, as in >>7149629
There's a clear logical connection between each step. No creative leaps of logic are required.[/spoiler]

>>7149629
If you want a more intuitive solution, notice that >>7149617
and then take the taylor series of the function exp(exp(x)) ,derive both sides three times and evaluate at x=0

>>7149585
It is convergent. Use the ratio test.

((k+1)^2)/(k!) times ((k-1)!)/(k^2) is

((k+1)^2)/(k^3)

Lim as k approaches infinity = 0, therefore convergent

>>7149646
can someone tell me why this is stupid?

>>7151820
The harmonic series diverges.

>>7151821
>harmonic series diverges
but wasn't there a thing like "anything that gets smaller faster than 1/k converges"

>>7149585
I'd use the ratio test if n goes to infinity

>>7150758
sum to n over k of 1/k is not convergent when n goes to infinity.

>>7151836
For any a>1, the sum over k of 1/k^a converges.

>>7151821
>>7151836
>>7151847
>>7151850

i got it
even 1/(k+1) is smaller than 1/k after all, and neither are convergent