series and sequences problem

[real analyst]

1. The problem statement, all variables and given/known data

I need to find three mathematical series, for the following patterns

1) Sn= 1/n if n is odd, or 1 of n is even.

2) Sn= 0,1,0,.5,1,0,1/3,2/3,0,1/4........

3) Find a sequence (Sk) which is a subset of the natural numbers in which every positive integer appears infinitely many times......that is, for each n in the Natural numbers, Sk=n for infinately many k's. any ideas???????????????????

2. Relevant equations

I guess they're arent any relevant equations to these. I guess yuo jsut have to get the pattern

3. The attempt at a solution

well, I know the first two are similar and thats about all I know. they're not p-series, geometric, alternating, or anything like that so I'm kind of stuck

The third one, well, I don't even kow what it means honestly, so can you at least give me a hint?? thanks.

[foxjwill]

For 1), try thinking along the lines of


(-1)^n =
\begin{cases}
-1 & n\mbox{ is odd}\\
1 & n\mbox{ is even}
\end{cases}


For 2), try thinking along the lines of the periodicity of \sin(x).

For 3), I'll have to refer you to the Wikipedia article on Hilbert's Paradox (http://en.wikipedia.org/wiki/Hilbert's_paradox).