/sci/ - Science & Math

The Game.

it converges to 2 you idiot

if 2+2 didn't equal 4, what would it equal?

>>3445056
>>3445056
Are you certain?

1 + 1/2 + 1/3 + 1/4 + 1/5 = ~2.08

5

Do you mean the converted function of the summation for values not infinity?

>>3445056
That would be 1 + 1/2 + 1/4 + 1/8..

If God didn't exist, then why would he exist?

Christians: 1
Athiests: checkmate

Are you talking about like how if you have a geometric series with r>1 it clearly diverges, but if you use the formula SUM=1/(1- r) anyway?

>>3445084
I'm sorry, I don't know what this means.

>>3445093
That same kind of thinking outside the box, yeah. There's several standard ways (Cesaro, Abel, etc.)

When you're adding up fractions inside a demoninator, there's a shortcut. For example, 1 / (1/a + 1/b) = ab/(a+b). So if S is the hypothetical sum of the harmonic series, you could use that shortcut to say that 1/S = (all natural numbers multiplied together) / (all natural numbers added together). In other words, S = (1+2+3+4+...) / (1*2*3*4*...)

Something called "zeta function regularization" can be used to sum 1+2+3+4+... to -1/12. But as for 1*2*3*4..., I have no idea.

It would not equal anything. It VERY slowly grows to infinity.

even 1/x^1.00000000000001 converges

>>3445106
Well, blindly chugging through the math with no regard for the concepts gets you things like string theory. Which could be good or bad, depending.
I'm not familiar with exactly how you prove the geometric sum formula, but I'm fairly certain the assumption that r<1 is explicity neccesary (meaning applying it to the wrong situations renders it unverified and illogical). Although I do remember reading in Penrose's "The Road to Reality" that such nonsesical sums may make sense when "treated carefully."
As to the harmonic series, I have no idea, but I've never heard of any of those methods you mentioned. I'll have to look into that.

How would one write a python program to calculate this for say, 100 iterations? It's been too long since I've used python, I can't remember how to do it. I know it's really simple though, I just can't get it to work.

>>3445132
Well, I'm asking what would happen if it didn't diverge. I'll admit this is rather poorly worded, but I can't resist trying to troll aspies even when I'm curious about something. What I'm really interested in is whether other summation methods would give a finite result, and what it would be.

>>3445208
I'm not sure that just picking a number like 100 out of the air would give much insight. The sum would eventually get arbitrarily large with enough iterations.

>>3445130
You could reguard S as the function
<div class="math">2 \frac{x!}{x(x+1)}</div>
by combining the ideas of factorials and triangular numbers. That should give you the sum of the first x harmonic numbers.
Just find the limit as that goes to infinity with L'Hospital's rule (I don't know how to differentiate the factorial though... You might have o get the gamma function involved.

>>3445235
Wait, something's wrong there. Bear with me; I'm doing this all in my head/on an iPhone.

>>3445130
Ah, the problem is that shorcut only works for a and b, not for a, b, c, etc. You can only sum two numbers.

>>3445208
Same person, just figured it out. I'm really a noob in python, but I was able to get it to work if I used one decimal place, like 1.0 instead of 1.

>>3445228
Thanks, I didn't realize that at first until I ran the program. I was thinking of the convergent series that went to 2.

>>3445289
Just physically adding the numbers will show you that they diverge, which is really the only good answer to OP's question.

>>3445362
>only good answer
I wouldn't be too sure of that. Coming up with finite values for divergent series can be quite useful.
http://en.wikipedia.org/wiki/Regularization_(physics)

>>3445289
not thisfag, but here's the code too, if anyone wants to play with it...
def Harmonic(n):
# does 'n' iterations of the Harmonic. ie:
# Harmonic(n) = 1/1 + 1/2 + 1/3 + ... + 1/n
SUM = 0.0
for k in range(1, n +1):
SUM += 1.0/k
return SUM
(pic related)

>>3445130
yeah this guy is right: >>3445283
for a,b,c, it becomes:
<span class="math"> \frac {1} { \frac {1} {a} + \frac {1} {b} + \frac {1} {c} } = \frac {abc} {ab + ac + bc}[/spoiler]

and for <span class="math">n[/spoiler] number of variables, it becomes:
<div class="math"> \frac {1} { \frac {1} {x_1} + \frac {1} {x_2} + \frac {1} {x_3} + ... + \frac {1} {x_n} } = \frac { \prod_{k=1}^n x_k } { \sum_{k=1}^n \left ( \prod_{j=1}^n \frac {x_j} {x_k} \right ) } </div>
(wow i hope that all works)


also, just on a note on this whole topic. i've seen sums like <span class="math">1 + 2 + 3 + ... = - \frac {1} {12}[/spoiler], using different "summation methods" like i've read here. but i never understood why there are different summation methods. or rather, why they are valid. i mean, isn't there only one good summation method: to just figgin' sum the gorram thing?
i would be very grateful if someone could tell me what are these other methods useful for / why do they exist?

(samefag bump? >>3445408)
also, just simplifying
<div class="math">\frac {1} { \sum_{k=1}^n \frac {1} {x_k} } = \frac { \prod_{k=1}^n x_k } { \sum_{k=1}^n \left ( \prod_{j=1}^n \frac {x_j} {x_k} \right ) }</div>