Analysis is easy

Let S = { \frac{x-2}{x+3} \mid x \in \Re, x \geq 0 }

Prove that Sup(S) = 1 and Inf(S) = -\frac{2}{3}

It's easy to see it's true but how to prove it fairly rigorously?

For any e > 0 you exhibit an x in R such that x-2/x+3 > 1-e

and similarly a y in R such that y-2/y+3 < -2/3 + e

That is if you know that 1 is an upper bound and -2/3 is a lower bound already, which you should also show, but that's basic.

so for example:

let x = n a natural number

then n-2/n+3 > n-2/n = 1 - 2/n  so that's Sup(S) done, as obviously x-2 < x+3 => x-2/x+3 < 1

Do the rest yourself

Thanks.

Another one for anyone who's bored.

Prove \frac{4^{k}}{k!} \rightarrow 0 as n \rightarrow \infty

>>3
That n should be a k sorry.

>>3
This stuff is pretty simple, what are you having trouble with?
Is it merely how to show the things "rigorously"?

In this case there's quite a few ways to do it.

I'd probably just say

4^k/k! < 4/k*(4^k-1/(k-1)!) < 4/8*(4^k-1/(k-1)!) for k>8

So if you let A_k = 4^k/k! you can see that A_k < 1/2*A_(k-1)


so as k-> infinity A_k < 1/2^n*A_8 -> 0 so done.

Obviously that could be tidied up, wanted to walk you through it

>>5

In the final line obvious n should be (k-8) or something similar, you get the idea.

>>5
You're right, this stuff is easy. I think I'm reading into the questions too much and thinking I have to do more than I need to. Thanks for your answer though, it's not quite how I would have done it.

hey, im a newfag in high school calculus. Excuse my ignorance, but what kind of math are you chaps doing here? I'm intrigued

>>8
http://en.wikipedia.org/wiki/Real_analysis

>>9
Don't bother read that.  Real analysis is just like calculus, only more boring.  Complex analysis is where it starts to get (slightly) interesting.

http://en.wikipedia.org/wiki/Complex_analysis

>>10
I found this interesting.
http://en.wikipedia.org/wiki/Basel_problem

The sum of the series 1/n^3 is unknown.

fourier series are more interesting, at this point you're just re-doing calculus

Personally I enjoyed the courses on real analysis more than complex, but I think functional will be better.

>>11
Oh and, it's not. Your link refers to the sum of 1/n^2, which is known to be pi/6.

And the sum on 1/n^3 is apéry's constant, the value of the riemann zeta function evaluated at 3.

It depends what you mean by known really.
It's been proved to be irrational, the fact that it has no nice closed form representation in terms of simple functions may just imply that none exist.

>>2
n-2/n+3 > n-2/n
wat

>>15
Fuck it, you get the idea. I hate typing maths anyway.