We already know the constant series diverges if a ≠ 0.
However, looking at the graph can help the intuition. Here's a 2-D graph of the constant series:
Putting all the little rectangles together creates an infinitely long rectangle of height a > 0. The area of such a rectangle must be infinite. Since the sum of the constant series is the area of that rectangle,
is infinite - in other words, the series diverges.
At long last, we can give a proof that the harmonic series diverges (even though its terms converge to 0).