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Two Concise Proofs of Harmonic Series Divergence

Plus the area under a curve without calculus.

The Harmonic Series provides excellent fodder for one studying infinite series. Let us dissect its infinite divergence. We will take two different approaches. First, a proof by contradiction.

Assuming the series converges (which it doesn’t, but when has that ever stopped us?) we represent the sum:

We can dice up the series and reassemble:

where

and

Before we go any further, a caveat. We can’t always get away with slicing up an infinite series. Take as an example:

Depending upon how I slice it, this series has a value of 0 or 1. It diverges: the partial sums alternate between 0 and 1 ad infinitum. It isn’t honing in on a single value.

There’s your key. We can only split up a convergent series. What’s more, the series must be absolutely convergent. Absolute convergence means the series converges even when we take the absolute value of each term. Only…