Llubanipo

2022-06-22

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 ${H}_{x}^{(k)}=\sum _{n=1}^{x}\frac{1}{{n}^{k}}$, and a harmonic number operating only on odd denominators ${G}_{x}=\sum _{n=1}^{x}\frac{1}{2n-1}$.

Clearly, there exist values of x,k such that ${G}_{x}<{H}_{x}^{(k)}$ and values such that ${H}_{x}^{(k)}<{G}_{x}$. Thus there exists a value $c={G}_{{x}_{0}}$ such that

${G}_{{x}_{0}}=c<{H}_{{x}_{0}}^{(k)}=\sum _{n=1}^{x}\frac{1}{{n}^{k}}$

and

${H}_{{x}_{0}+2}^{(k)}<{G}_{{x}_{0}+2}=c+\frac{1}{2{x}_{0}+1}+\frac{1}{2{x}_{0}+3}$

or

${H}_{{x}_{0}+2}^{(k)}-c<\frac{1}{2{x}_{0}+1}+\frac{1}{2{x}_{0}+3}$

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 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 ${G}_{x}=\sum _{n=1}^{x}\frac{1}{2n-1}$.

Clearly, there exist values of x,k such that ${G}_{x}<{H}_{x}^{(k)}$ and values such that ${H}_{x}^{(k)}<{G}_{x}$. Thus there exists a value $c={G}_{{x}_{0}}$ such that

${G}_{{x}_{0}}=c<{H}_{{x}_{0}}^{(k)}=\sum _{n=1}^{x}\frac{1}{{n}^{k}}$

and

${H}_{{x}_{0}+2}^{(k)}<{G}_{{x}_{0}+2}=c+\frac{1}{2{x}_{0}+1}+\frac{1}{2{x}_{0}+3}$

or

${H}_{{x}_{0}+2}^{(k)}-c<\frac{1}{2{x}_{0}+1}+\frac{1}{2{x}_{0}+3}$

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 $c,k$?

jarakapak7

Beginner2022-06-23Added 14 answers

We can approximate the sums by the integrals and then to deal with the resulting functions. For instance, for $k<1$,

${H}^{(k)}(x)\approx {\int}_{1}^{x}\frac{1}{{t}^{k}}dt=\frac{{t}^{1-k}}{1-k}{|}_{1}^{x}=\frac{1}{1-k}({x}^{1-k}-1).$

For $k=1$ and natural $x$,

${H}^{(1)}(x)\sim \mathrm{ln}x+\gamma +\frac{1}{2x}-\frac{1}{12{x}^{2}}+\frac{1}{120{x}^{4}}-\dots ,$

where $\gamma \approx 0.5772156649$ is the Euler–Mascheroni constant. For $k>1$ when $x$ tends to infinity, the sequence ${H}_{x}^{(k)}$ converges to Riemann zeta function $\zeta (k)$.

Similarly we have

$G(x)\approx {\int}_{1}^{x}\frac{1}{2t-1}dt=\frac{1}{2}\mathrm{ln}(2x-1).$

${H}^{(k)}(x)\approx {\int}_{1}^{x}\frac{1}{{t}^{k}}dt=\frac{{t}^{1-k}}{1-k}{|}_{1}^{x}=\frac{1}{1-k}({x}^{1-k}-1).$

For $k=1$ and natural $x$,

${H}^{(1)}(x)\sim \mathrm{ln}x+\gamma +\frac{1}{2x}-\frac{1}{12{x}^{2}}+\frac{1}{120{x}^{4}}-\dots ,$

where $\gamma \approx 0.5772156649$ is the Euler–Mascheroni constant. For $k>1$ when $x$ tends to infinity, the sequence ${H}_{x}^{(k)}$ converges to Riemann zeta function $\zeta (k)$.

Similarly we have

$G(x)\approx {\int}_{1}^{x}\frac{1}{2t-1}dt=\frac{1}{2}\mathrm{ln}(2x-1).$