/sci/ - Science & Math

>>12253099
Don't get caught up in the opinions of some angry undergrad.

E.g. you have for |x| < 1, you have

[math] \sum_{k=0}^\infty x^k = \dfrac{1}{1-x} [/math]

and so with [math] x = -1+\epsilon [/math], for positive [math]\epsilon[/math], you find

[math] \sum_{k=0}^\infty (-1)^k \cdot (1 - \epsilon)^k = \dfrac{1}{2-\epsilon} [/math]

So from this we e.g. know that
[math] \sum_{k=0}^\infty (-0.99999999)^k [/math]
is something extremely close to [math]1/2[/math].
But note that your calulator will very quickly give you numerical precision errors with a number like that.

While we can't plug in [math] \epsilon = 0 [/math] in clasical analysis to obtain

[math] \sum_{k=0}^\infty (-1)^k = \dfrac{1}{2} [/math],

it's not like the value unmotivated either.

>So why does anyone care?
You can see math as inferring from axioms. Some theories are useful (e.g. in geometry), some are not. Summation methods for sums that are divergent in classical analysis aren't extremely useful, empirically speaking. But neither is computing cohomology groups of 17 dimensional spheres. Some people care because it's also math.

>>12253099
Don't get caught up in the opinions of some angry undergrad.

E.g. you have for |x| < 1, you have

[math] \sum_{k=0}^\infty x^k \cdot = \dfrac{1}{1-x} [/math]

and so with [math] x = -1+\epsilon [/math], you find

[math] \sum_{k=0}^\infty (-1)^k \cdot (1 - \epsilon)^k = \dfrac{1}{2-\epsilon} [/math]

So from this we e.g. know that [math] \sum_{k=0}^\infty (-0.99999999)^k \cdot [/math] is something extremely close to [math]1/2[/math].
(But note that your calulator will very quickly give you numerical precision errors with a number like that)

While we can't plug in [math] \epsilon = 0 [/math] in clasical analysis to obtain

[math] \sum_{k=0}^\infty (-1)^k = \dfrac{1}{2} [/math],

it's not like the value unmotivated either.

>So why does anyone care?
You can see math as inferring from axioms. Some theories are useful (e.g. in geometry), some are not. Summation methods for sums that are divergent in classical analysis aren't extremely useful, empirically speaking. But neither is computing cohomology groups of 17 dimensional spheres. Some people care because it's also math.