Math Is Fun Forum

Ok, I thought that {0,n} was {0,1,...,n}

uhmm i'm trying to figure out how do you see that f goes to a y0... let's see:

so the c(n) part converges and the x part of course does. Is there a easiest way to see that?

Thank you, this remark:
"Peano (1889) observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent."
opened my eyes

Hi there! These are very simple and smart proofs. I suggest you also to give a look to:

http://farside.ph.utexas.edu/euclid/Elements.pdf
Book1, Prop.48, Prop.49

the second one (if i remember good) is the Inverse theorem (i.e. if a triangle is such that c²=a²+b², then it's rectangle).

I had a quick glance and it is very obscure to me; i'll try to read it later, thank you.

cool!

I have to admit that at first sight this looked funny; but after being (maybe) less superficial i'm seeing a meaning behind this:
look it geometrically (i write polar coordinates for complex numbers)...

the (first) square root for -1 is    (1,pi/2)       (midnight)
the (first) 3rd root for  -1        (1,pi/3)          (one o'clock)
the (first) 4th root for -1 is    (1,pi/4)            (half past one)
.....
.....                                                             (...some time passes...)
.....
the (first) nth root for -1 tends to (1,0)         (almost three o' clock)

so it seems to me that your limit is what the first nth root of (-1) tends to.

EDIT: I want to add something:

where k=0,1,2...,n-1. In particular, the integer part of (n+1)/2 (which is n/2 if n is even and (n+1)/2 if odd) belongs to the list of k's;
If we accept your and my proceeding then we get:

(where i put n/2 or n+1/2 as k)

so one of us (or eventually both ) must be wrong.

hi mitu, I would say it does not exist if your succession is from N to R;
here's a short proof of non-existence:
if LIM[a(n)]=L, then for all a(n(k)) LIM[a(n(k))]=L
you see that LIM[a(2k)]!=L since a(2k) is not defined for each k. But maybe someone would argue that for each n in dom(a(n)) a(n)=-1, so LIMa(n)=-1... i see it just as a formal problem, maybe someone can be more precise.

While writing my post i realized that if your succession is from N to C it is not even a function, so i don't know if it has any meaning to talk about limit...

solved, correct proposition is: