$ \sum_{k=2}^{\infty} \left( \frac{1}{k-1} - \frac{1}{k} \right) = 1$

Hint:

enter image description here

hope you can take it from here.

As @CarstenS pointed out the picture shown here is somehow a more correct hint


Note that

$$\begin{align} \sum_{k=2}^K\left(\frac{1}{k-1}-\frac1k\right)&=\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\left(\frac13-\frac14\right)+\cdots +\left(\frac{1}{K-1}-\frac{1}{K}\right)\\\\ &=1-\frac1K \tag 1 \end{align}$$

More formally, note that if $S_K=\sum_{k=2}^K\left(\frac{1}{k-1}-\frac1k\right)$, then $S_{K+1}=S_K+\frac{1}{K}-\frac{1}{K+1}$. So, we can use induction to prove the result given by $(1)$.

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