Exponential functions are characterized by the property that the graph of the derivative is the graph of scaled by the factor of in the vertical direction: . We also have for the sake of normalization.
What happens if we replace “vertical” by “horizontal”? That is, let be the function such that , still with for normalization. Clearly, . Let’s consider from now on. Assuming we can express as a power series , the comparison of and yields the recurrence relation . This relation leads to an explicit formula: where . That is,
The behavior of , and in the vicinity of is shown below.
A few questions come up. Is strictly increasing? Does it have a finite limit at ? If so, is this limit negative? If so, where does cross the -axis? What is its rate of growth as ? Does the number have any significance, considering that ?
First of all: if does become negative at some point then its derivative also becomes negative further to the left, which can make rise to positive values again, and then the process will probably repeat as shown below. Also, is negative at each point of local minimum, and positive at each point of local maximum. This is because both the function and its derivative are positive at , and moving to the left, the derivative cannot change the sign until the function itself changes the sign.
When is close to , the oscillating pattern takes longer to develop: here it is with . Note the vertical scale: these are very small oscillations, which is why this plot does not extend to the zero mark.
For any the alternating series estimate gives when , hence for . It follows that is strictly increasing when . We have
Since is decreasing for , the alternating series estimate applies and shows that
So, when we have and therefore there is a unique point where . Specifically for , this root is . Its significance is that is the critical point closest to , meaning that is the largest interval of monotonicity of .
In general it is not true that . Indeed, uniformly on bounded sets as . I do not have a proof that has a real root for every .
When , the sequence of denominators is A011266 in OEIS (it is related to counting the evil and odious numbers). But the sum of its reciprocals, , did not show up anywhere.