I wanted an example of a function that behaves mostly like (the product is bounded between two positive constants), but such that does not have a limit as .
The first thing that comes to mind is , but this function does not look very much like .
Then I tried , recalling an example from Linear Approximation and Differentiability. It worked well:
In fact, it worked much better than I expected. Not only if of constant sign, but so are and . Indeed,
is always negative,
is always positive,
is always negative. The sign becomes less obvious with the fourth derivative,
because the triangle inequality isn’t conclusive now. But the amplitude of is , and .
So, it seems that is completely monotone, meaning that for all and for all . But we already saw that this sign pattern can break after many steps. So let’s check carefully.
Direct calculation yields the neat identity
With its help, the process of differentiating the function can be encoded as follows: , , then and . The presence of is disconcerting because the harmonic series diverges. But orthogonality helps: the added vector is orthogonal to .
The above example, rewritten as , corresponds to starting with . I calculated and plotted iterations: the points are joined by piecewise linear curve.
The total length of this curve is infinite, since the harmonic series diverges. The question is, does it stay within the unit disk? Let’s find out. By the above recursion,
Hence, the squared magnitude of will always be less than
with being . The infinite product evaluates to (explained here), and thus the polygonal spiral stays within the disk of radius . In conclusion,
where the trigonometric function has amplitude strictly less than . Since the expression on the right is positive, is completely monotone.
The plot was generated in Sage using the code below.
a,b,c,d = var('a b c d') a = 0 b = 1/2 l = [(a,b)] for k in range(1,10000): c = a-b/k d = b+a/k l.append((c,d)) a = c b = d show(line(l),aspect_ratio=1)