The classic Gauss trick 😉
$$1 + 2 + 3 + \cdots + 100 = \,?$$

🍕 You eat half a pizza, then half of what is left, then half of that — forever:
$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$$

How many whole pizzas do you eat?

🔗 The collapsing chain — compute as a fraction:
$$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\cdots+\frac{1}{99\times 100}$$