Infinity in Mathematics (Calculus and Series)

[kurt.physics]

Could some one please explain to me 2 things

1) I have seen integrals that are between 0 and ∞ and also between -∞ and ∞. What does this mean

2) I have also seen sigma series (∑) between n=1 and ∞. What doe this mean

Thanks heaps

 

[CompuChip]

Actually, an expression like
$$\int_0^\infty f(x) \, \mathrm{d}x$$
is just a shorthand notation for
$$\lim_{a \to \infty} \int_0^a f(x) \, \mathrm{d}x.$$
Likewise,
$$\int_{-\infty}^\infty f(x) \, \mathrm{d}x = \lim_{a \to -\infty} \int_a^0 f(x) \, \mathrm{d}x + \lim_{b \to \infty} \int_0^b f(x) \, \mathrm{d}x$$
and
$$\sum_{n = 1}^\infty a_n = \lim_{N \to \infty} \sum_{n = 1}^N a_n.$$

 

[kurt.physics]

Thanks CompuChip, but i don't understand, how do they exist, what do they literally mean, could you give a example please.

Thanks

 

[matt grime]

I think we need to find out what you think an integral, and a limit are, since the above seem self explanatory.

 

[CompuChip]

For example, suppose we want to evaluate
$$\int_0^\infty e^{-x} \, \mathrm{d}x.$$
Then we first calculate
$$\int_0^a e^{-x} \, \mathrm{d}x = \left. -e^{-x} \right|_{x = 0}^a = -e^{-a} - (-e^{-0}) = 1 - e^{-a}$$.
Now take the limit:
$$\int_0^\infty e^{-x} \, \mathrm{d}x = \lim_{a \to \infty} \int_0^a e^{-x} \, \mathrm{d}x = \lim_{a \to \infty} 1 - e^{-a} = 1 - \lim_{a \to \infty} e^{-a} = 1 - 0 = 1.$$

That's all there is to it... which part didn't you understand?

 

[kurt.physics]

Thanks, it makes perfect sense now :)