Spivak Calculus chapter 2 problem 6 - what's the intuition behind this technique?

Question

The following technique could be used to derive the closed-form formulas for $1^a+...+n^a$ which I have verified so far for $a = 2,3,4$.

1. Let's assume $a=2$, expand $(k+1)^{a+1=3}$.
2. We now have $(k+1)^3= k^3+3k^2+3k+1.$
3. Bring over the $k^3$ producing: $(k+1)^3 -k^3=3k^2+3k+1.$
4. Take a sum of this equation for $k=1...n$.
5. That yields: $(n+1)^3-1=3[1^2+...+n^2]+3[1+...+n]+n$.
6. Set $S=[1^2+...+n^2]$ and solve for $S$ which yields the familiar: $n(n+1)/2$

The problem is now I need to use this technique to find the closed-form formula for: $\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{n(n+1)}$

Two basic approaches came to mind:

1. Transform this input in such a way that it takes the familiar form from the example above. This could even entail breaking the input into pieces, running through the technique for each, and then assembling the solutions. Seems nice in theory, but I didn't make any progress.
2. Determine if this technique is really a "sub-class of some base-class". In other words, is there an essence to this technique that could be distilled from which a different "sub-class" tailored to processing the input could be built? In other words, should I try to change the technique from being dependent on the binomial expansion $(k+1)^a$? If so, this unlike approach (1) requires an understanding of how this technique really works, which brings me to my question. How could I understand why this works in the first place?