# How to determine bounds on one variable in a system of inequalities?

I came across this problem whilst exploring the asymptotic behaviour (or not) of different generalised harmonic numbers. 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 $$H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$$, and a harmonic number operating only on odd denominators $$G_x=\sum_{n=1}^x \frac{1}{2 n-1}$$.

Clearly, there exist values of $$x,k$$ such that $$G_x and values such that $$H_x^{(k)}. Thus there exists a value $$c=G_{x_0}$$ such that

$$G_{x_0}=c and $$H_{{x_0}+2}^{(k)} or $$H_{{x_0}+2}^{(k)}-c<\frac{1}{2x_0+1}+\frac{1}{2x_0+3}$$

The values of $$c,x_0,k$$ are obviously co-dependent. I am searching for a way to solve for $$x_0$$ 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 $$G$$ and I want to be able to explore how to solve such problems generally, for whatever pattern of $$G$$ I choose (provided it's formulable!).

Algebraically, how do I put bounds on $$x_0$$ in terms of $$c,k$$?

(I am an amateur, so I need a fair amount of hand-holding, hence the bounty.)

• Hi @Saad. Humble apologies - please see revision. Commented May 4, 2020 at 13:01
• It does. My bad - a hangover from the over-terse prec=vious question, now fixed. Commented May 4, 2020 at 16:31

I am interested in how to approach this algebraically rather than numerically. This is a single simple example of $$G$$ and I want to be able to explore how to solve such problems generally, for whatever pattern of $$G$$ I choose (provided it's formulable!).
We can approximate the sums by the integrals and then to deal with the resulting functions. For instance, for $$k<1$$, $$H^{(k)}(x)\approx \int_{1}^{x} \frac 1{t^k} dt=\frac{t^{1-k}}{1-k}{\Huge |}^{x}_1=\frac{1}{1-k}\left(x^{1-k}-1\right).$$ For $$k=1$$ and natural $$x$$, according to Wikipedia, $$H^{(1)}(x)\sim\ln x+\gamma+\frac{1}{2x}- \frac{1}{12x^2}+\frac{1}{120x^4}-\dots,$$
where $$\gamma\approx 0.5772156649$$ is the Euler–Mascheroni constant. For $$k>1$$ when $$x$$ tends to infinity, the sequence $$H^{(k)}_x$$ converges to Riemann zeta function $$\zeta(k)$$.
Similarly we have $$G(x)\approx \int_1^{x}\frac {1}{2t-1}dt=\frac 12\ln (2x-1).$$
• Hi @Alex. How would I isolate the required value of $x$ from this? Commented May 6, 2020 at 10:52
• @RichardBurke-Ward A good question. I have to confess that I didn’t understand the details of you question. But I guess that an analytic approach for bounds for $x_0$ should start from the above observations. Maybe somebody more clever than me would be able to finish it. Commented May 6, 2020 at 17:31