Value of an infinite series I

Question:
Consider the following infinite series:
\[S_n = 2-\frac{3}{2}+\frac{3}{2}-\frac{4}{3}+\frac{4}{3}-\frac{5}{4}+ \cdots\]
Find the value(s) when $n \rightarrow \infty$.

Answer:
This series can be generalized as follows:
\[
S_n = \sum^{\infty}_{n=1}\left(\frac{n+1}{n}-\frac{n+2}{n+1}\right)
\]
The value depends on the number of terms. When $n$ is even, the last term should be $-\frac{n+2}{n+1}$. Namely,
\begin{eqnarray}
S_n &=& 2-\frac{3}{2}+\frac{3}{2}-\frac{4}{3}+\frac{4}{3}-\frac{5}{4}+ \cdots +\frac{n+1}{n}-\frac{n+2}{n+1}  \\
   &=& 2 - \frac{n+2}{n+1}
\end{eqnarray}
Except the first and last terms, all of them are cennceled out. Therefore, the value of the infinite series is
\[
\lim_{n \rightarrow \infty}S_n = \lim_{n \rightarrow \infty}\left(2-\frac{1+\frac{2}{n}}{1+\frac{1}{n}}\right)=1
\]
When $n$ is odd, the series ends up with $+\frac{n+1}{n}$. Namely,
\begin{eqnarray}
S_n &=& 2-\frac{3}{2}+\frac{3}{2}-\frac{4}{3}+\frac{4}{3}- \cdots -\frac{n+1}{n}+\frac{n+1}{n}  \\
   &=& 2
\end{eqnarray}
Thus we have the value.
\[
\lim_{n \rightarrow \infty}S_n = 2
\]
The value oscillates between 1 and 2.

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