if .999,,,,=1 does .999,,,,+.999,,,= 2 or 1.999,,,8?

[Mathochist]

@Ellis Dee

Doesn’t look like there’s going to be a big crunch in our universe’s future. All evidence points to the contrary: the eventual “heat-death” of maximum entropy, long after the last star has burned out and the last black hole has evaporated, one particle at a time.

I was bummed out when I heard about this discovery. It appears that the universe will have active stars for as much of its lifespan as the earth has had humans in its history up to now. An almost negligably small fraction.

Don’t become a mathematician then. It’s provable that all the interesting stuff never happens.

[Hyperelastic]

Does

m

\ 0.99999… = m
/

n=1
hold for m=infinity?

[Mathochist]

@Hyperelastic

Does

m

\ 0.99999… = m
/

n=1
hold for m=infinity?

Well first let’s write down something that makes sense.

lim[sub]m->infinity[/sub] ?[sub]n=1[/sub][sup]m[/sup] lim[sub]l->infinity[/sub] ?[sub]k=1[/sub][sup]l[/sup] 9/10[sup]k[/sup]

Since the outer sum doesn’t involve l we can pull the limit outside

lim[sub]m->infinity[/sub] lim[sub]l->infinity[/sub] ?[sub]n=1[/sub][sup]m[/sup] ?[sub]k=1[/sub][sup]l[/sup] 9/10[sup]k[/sup]

Similarly, we can exchange order of summation

lim[sub]m->infinity[/sub] lim[sub]l->infinity[/sub] ?[sub]k=1[/sub][sup]l[/sup] ?[sub]n=1[/sub][sup]m[/sup] 9/10[sup]k[/sup]

And the limit over m can be pulled inside

lim[sub]l->infinity[/sub] ?[sub]k=1[/sub][sup]l[/sup] lim[sub]m->infinity[/sub] ?[sub]n=1[/sub][sup]m[/sup] 9/10[sup]k[/sup]

Now the inner sum can be taken

lim[sub]l->infinity[/sub] ?[sub]k=1[/sub][sup]l[/sup] lim[sub]m->infinity[/sub] 9m/10[sup]k[/sup]

And when we take the inner limit, that diverges. Since all the terms in the remaining sum are positive divergent, the whole sum is as well. That’s about as much sense as anyone can make of that expression.

[mangeorge]

@Ellis Dee

Doesn’t look like there’s going to be a big crunch in our universe’s future. All evidence points to the contrary: the eventual “heat-death” of maximum entropy, long after the last star has burned out and the last black hole has evaporated, one particle at a time.

I was bummed out when I heard about this discovery. It appears that the universe will have active stars for as much of its lifespan as the earth has had humans in its history up to now. An almost negligably small fraction.

The Universe is going to… Fizzle???
Sniff.