Series Expansion/Taylor Series Help

Okay, I am in dire need of help.

How do I know that sin(x) is equal to x-((x^3)/6)+......+(-1)^n((x^(2n+1))/(2n+1)!)+x^(2n+1)E(x), (where E is epsilon)?

Same goes for cos(x), arctan(x), and all the other functions that can be represented by the sum of a series(I think that's how you say it).

Now, for the actual math.

How do I calculate the limited series expansion for f(x)=(sqrt(1+x))(sin(x)), (where sqrt is the square root of) to the order of 2 at 0? How do I calculate to the order 4 at 1 the function f(x)=(ln(x))/x^2?

I really need help!

Also, I don't get what this notation means: h: J -> I, y ->h (y), where -> is an arrow. What does that mean? Sometimes I see it like this: f : I -> J or f : I -> R, with R standing for all real numbers. What is that?

Thanks.