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 ) = ∑ n = 1 x 1 n k , and a harmonic number operating only on odd denominators G x = ∑ n = 1 x 1 2 n − 1 .Clearly, there exist values of x,k such that G x &lt; H x ( k ) and values such that H x ( k ) &lt; G x . Thus there exists a value c = G x 0 such that G x 0 = c &lt; H x 0 ( k ) = ∑ n = 1 x 1 n k and H x 0 + 2 ( k ) &lt; G x 0 + 2 = c + 1 2 x 0 + 1 + 1 2 x 0 + 3 or H x 0 + 2 ( k ) − c &lt; 1 2 x 0 + 1 + 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?

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How to determine bounds on one variable in a system of inequalities?I am interested in the point of &#039;cross-over&#039; 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      )        =      ∑          n      =      1        x        1          n      k      , and a harmonic number operating only on odd denominators       G    x    =      ∑          n      =      1        x        1          2      n      −      1      .Clearly, there exist values of x,k such that       G    x    &amp;lt;      H    x          (      k      )       and values such that       H    x          (      k      )        &amp;lt;      G    x  . Thus there exists a value   c  =      G            x      0       such that      G            x      0        =  c  &amp;lt;      H            x      0              (      k      )        =      ∑          n      =      1        x        1          n      k      and      H                      x        0            +      2              (      k      )        &amp;lt;      G                      x        0            +      2        =  c  +      1          2              x        0            +      1        +      1          2              x        0            +      3      or      H                      x        0            +      2              (      k      )        −  c  &amp;lt;      1          2              x        0            +      1        +      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&#039;s formulable!).Algebraically, how do I put bounds on       x    0   in terms of   c  ,  k?

jarakapak7 · Accepted Answer

We can approximate the sums by the integrals and then to deal with the resulting functions. For instance, for   k  &amp;lt;  1,      H          (      k      )        (  x  )  ≈      ∫      1      x        1          t      k        d  t  =            t              1        −        k                    1      −      k                        |        1      x    =      1          1      −      k            (          x              1        −        k              −    1    )    .For   k  =  1 and natural    x,      H          (      1      )        (  x  )  ∼  ln  ⁡  x  +  γ  +      1          2      x        −      1          12              x        2              +      1          120              x        4              −  …  ,where   γ  ≈  0.5772156649 is the Euler–Mascheroni constant. For   k  &amp;gt;  1 when   x tends to infinity, the sequence       H    x          (      k      )       converges to Riemann zeta function   ζ  (  k  ).Similarly we have  G  (  x  )  ≈      ∫    1      x        1          2      t      −      1        d  t  =      1    2    ln  ⁡  (  2  x  −  1  )  .

How to determine bounds on one variable in a system of inequalities?I am interested in...

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