The function is odd and so to study the roots of we only need to study positive roots of which is what we’re going to do in this post. We begin with a basic fact.
Problem 1. Consider the function For an integer let be the interval Show that
i) is increasing on any interval which is in its domain
ii) the set of positive roots of is a sequence where
ii) Let be an integer. We have Also, the limit of as approaches from the left, is Since is continuous in the intermediate value theorem implies that for some Since, by i), is increasing on is the unique root of in
Finally, has no other positive roots because is positive on and negative on the interval for any integer (why ?).
Next, given integer we’re going to refine the basic result given in Problem 1, ii), by finding an interval which is much smaller than and still contains But first we need to prove something.
Problem 2. Show that the function
Solution. We find the first and the second derivatives of we have
So and have the same sign because both and are positive on the interval Thus the maximum of occurs at and so
Problem 3. Let be the sequence of positive roots of Show that
Solution. Let and We are done if we show that and Let
By Problem 2, is decreasing and thus for all In particular, i.e.
On the other hand, since for all and is decreasing, we have i.e.
and that gives
Example. For Problem 3 gives and the actual value of is approximately So the lower and upper bounds that Problem 3 gives for are not that bad.
Exercise. Let be the positive solutions of the equation
i) Show that and
ii) Show that the series is divergent.
iii) Let be the floor function. Is it true that for all