How to I find a formula for $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)}$ using the given method? The following is problem $6$ $(iii)$ from chapter $2$ of Spivak's Calculus:

The formula for $1^{2} + \cdots + n^{2}$ may be derived as follows. We begin with the formula $$ (k+1)^{3} - k^{3} = 3k^{2}+3k +1$$.
Writing this formula for $k=1, \cdots , n$ and adding
\begin{align*}
 2^{3}-1^{3}&=3\cdot1^{2} + 3\cdot1 + 1 \\
 3^{3}-2^{3}&=3\cdot2^{2} + 3\cdot2 + 1 \\
&\vdots\\
(n+1)^{3}-n^{3}&=3\cdot n^{2} + 3\cdot n +1 \\
\end{align*}
we obtain
  $$(n+1)^{3}-1 = 3[1^{2} + \cdots + n^{2}] + 3[1 + \cdots + n] + n.$$
Thus we can find $$\sum_{k=1}^n k^{2}$$ if we already know $$\sum_{k=1}^n k .$$ Use this method to find
$$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)}$$.

So originally, I attempted $$\sum_{k=1}^n \frac{1}{k} - \sum_{k=1}^n \frac{1}{k+1}.$$ This means that we have
$$(1 + \frac{1}{2} + \cdots + \frac{1}{n}) - (\frac{1}{2} + \frac{1}{3} + \cdots \frac{1}{n}+\frac{1}{n+1})$$
which gives us $1-\dfrac{1}{n+1}$. Although, I'm not sure if the method I've used suffices as the method required for the problem; that is, it's supposed to look similar to the example using $(k+1)^{3}$ above.

It is a perfectly fine answer. The above method which is being illustrated is known as Telescopic method of summation. You can read more about it here.
https://en.wikipedia.org/wiki/Telescoping_series
https://brilliant.org/wiki/telescoping-series/ 
