Problem source

Sketch the graph of the function $[x/N]$, for $0   <   x < 2N$, where 
the notation  $[y]$ means the integer part of $y$.
(Thus $[2.9] = 2$, \ $[4]=4$.) 
\begin{questionparts}
\item Prove that 
\[
\sum_{k=1}^{2N} (-1)^{[k/N]} k = 2N-N^2.
\]
\item Let
\[
S_N = \sum_{k=1}^{2N} (-1)^{[k/N]} 2^{-k}.
\]
Find  $S_N$ in terms of $N$ and determine the limit of $S_N$ as  $N\to\infty$.
\end{questionparts}