How to determine bounds on one variable in a system of inequalities?I am interested in...
How to determine bounds on one variable in a system of inequalities?
I am interested in the point of 'cross-over' between a generalised harmonic number where the denominator of the summand is raised to a power, and a non-exponential harmonic sum operating on some subset of the natural numbers.
For example, take the generalised harmonic number , and a harmonic number operating only on odd denominators .
Clearly, there exist values of x,k such that and values such that . Thus there exists a value such that
The values of are obviously co-dependent. I am searching for a way to solve for or at least put bounds on it.
I am interested in how to approach this algebraically rather than numerically. This is a single simple example of and I want to be able to explore how to solve such problems generally, for whatever pattern of I choose (provided it's formulable!).
Algebraically, how do I put bounds on in terms of ?