Before I do proofs via the 3-step mathematical induction, the classic Charlie Chaplin movie clip often comes to my mind. It nudges me to seek better ways. For example, to prove

,

Instead of applying the 3-step mathematical induction, let us look at Fig. 1.

Fig. 1

If *A* denotes the total area of the blue rectangles, Fig.1 shows

.

It also shows that

A < the area under the curve .

By the fundamental theorem of calculus,

.

Therefore,

,

i.e.,

.

This proof suggests another inequality:

.