Gfilui

This recent question is the motivation for this post.

Consider the following equation $$frac1x-1+frac1x-2+cdots+frac1x-k=frac1x-k-1$$ where $k>1$.

My claims:

1. There are $k$ solutions, all of which are real.




2. Let $x_min$ be the minimum value of these $k$ solutions. Then as $ktoinfty$, $x_min$ converges. (If it does, to what value does it converge?)




3. As $ktoinfty$, all of the solutions get closer and closer to an integer, which is bounded below. Furthermore, these integers will be $1, 2, 3, cdots, k-1, k+1$.

To see these patterns, I provide the solutions of $x$ below. I used W|A for $kge4$. The values in $colorbluetextblue$ are those of $x_min$.

$$beginarrayck&2&3&4&5&6\hline x&4.414&4.879&5.691&6.592&7.530\&colorblue1.585&2.652&3.686&4.701&5.722\&&colorblue1.468&2.545&3.588&4.615\&&&colorblue1.411&2.487&3.531\&&&&colorblue1.376&2.449\&&&&&colorblue1.352endarray$$

Also, when $k=2$, the polynomial in question is $x^2-6x+7$, and when $k=3$, it is $x^3-9x^2+24x-19$.

The reason why I think $x_min$ converges is because the difference between the current one and the previous gets smaller and smaller as $k$ increases.

Are my claims true?

Both your claims are true.

if you call
$$
f(x) = frac1x-1+frac1x-2+cdots+frac1x-k-frac1x-k-1
$$
then $f(1^+) = +infty$, $f(2^-) = -infty$ and $f$ is continuous in $(1,2)$, so it has a root in $(1,2)$.
The same you can say about $(2,3)$, $(3,4), cdots, (k-1,k)$, so there are at least $k-1$ real distinct roots. $f$ is also equivalent to a $k$-degree polynomial with the same root, but a $k$-degree polynomial with $k-1$ real roots has in reality $k$ real roots.

The last root lies in $(k+1,+infty)$, since $f(k+1^+) = -infty$ and $f(+infty) = +infty$.

The least root $x_min$ must lie in $(1,2)$, since $f(x)<0$ for every $x<1$. Moreover,
$$
f(x) = 0implies x = 1 + frac1frac1x-k-1-frac1x-2-cdots-frac1x-k
$$
and knowing $1<x<2$, we infer $frac1x-k-1>frac1x-2$ and
$$
1<x = 1 + frac1frac1x-k-1-frac1x-2-cdots-frac1x-k
< 1 - frac1frac1x-3+cdots+frac1x-kto 1
$$
so $x_min$ converges to $1$

About the third claim, notice that you may repeat the same argument for any root except the biggest. Let us say that $x_r$ is the $r-th$ root, with $r<k$, and we know that $r<x_r<r+1$.
$$
f(x_r) = 0implies x_r = r + frac1frac1x_r-k-1-frac1x_r-1-cdots-frac1x_r-k
$$
but $frac1x_r-k-1>frac1x_r-1$ holds, so
$$
r<x_r = r + frac1frac1x_r-k-1-frac1x_r-1-cdots-frac1x_r-k
< r - frac1frac1x_r-2+cdots+frac1x_r-kto r
$$
so $x_r$ converges to $r$.

For the biggest root, we know $k+1<x_k$ and
$$
f(x_k) = 0implies k+1 < x_k = k+1 + frac1frac1x_k-1+cdots+frac1x_k-k to k+1
$$

For the base case,
$$tag1f_2(x)=frac1x-1+frac1x-2-frac1x-3, $$
one readily verifies that there is a root in $(1,2)$ and a root $x^*$ in $(3,+infty)$.

If we multiply out the denominators of
$$f_k(x)=frac1x-1+frac1x-2+ldots+frac1x-k-frac1x-k-1,$$
we obtain the equation
$$tag2(x-1)(x-2)cdots(x-k-1)f_k(x)=0,$$
which is a polynomial of degree (at most) $k$, so we expect $k$ solutions, but some of these may be complex or repeated or happen to be among $1,2,ldots, k+1$ and thus not allowed for the original equation.
But $f_k(x)$
has simple poles with jumps from $-infty$ to $+infty$ at $1,2,3,ldots, k$, and a simple pole with jump from $+infty$ to $-infty$ at $k+1$, and is continuous otherwise. It follows that there is (at least) one real root in $(1,2)$, at least one in in $(2,3)$, etc. up to $(k-1,k)$, so there are at least $k-1$ distinct real roots.
Additionally, for $x>k+1$ and $kge2$, we have
$$f_k(x)ge f_2(x+k-2).$$
It follows that there is another real root between $k+1$ and $x^*+k-2$.
So indeed, we have $k$ distinct real roots.

From the aboive, the smallest root is always in $(1,2)$.
If follows from $f_k+1(x)>f_k(x)$ for $xin(1,2)$ and the fact that all $f_k$ are strictly decreasing there, that $x_min $ decreases with increasing $k$. As a decreasing bounded sequence, it does have a limit.

Considering that you look for the first zero of function
$$f(x)=sum_i=1^k frac 1x-i-frac1 x-k-1$$ which can write, using harmonic numbers,
$$f(x)=H_-x-H_k-x-frac1x-k-1$$ remove the asymptotes using
$$g(x)=(x-1)(x-2)f(x)=2x-3+(x-1)(x-2)left(H_2-x-H_k-x-frac1x-k-1 right)$$ You can approximate the solution using a Taylor expansion around $x=1$ and get
$$g(x)=-1+(x-1) left(-frac1k+psi ^(0)(k)+gamma
+1right)+Oleft((x-1)^2right)$$ Ignoring the higher order terms, this gives as an approximation
$$x_est=1+frackkleft(gamma +1+ psi ^(0)(k)right)-1$$ which seems to be "decent" (and, for sure, confirms your claims).
$$left(
beginarrayccc
k & x_est & x_sol \
2 & 1.66667 & 1.58579 \
3 & 1.46154 & 1.46791 \
4 & 1.38710 & 1.41082 \
5 & 1.34682 & 1.37605 \
6 & 1.32086 & 1.35209 \
7 & 1.30238 & 1.33430 \
8 & 1.28836 & 1.32040 \
9 & 1.27726 & 1.30914 \
10 & 1.26817 & 1.29976 \
11 & 1.26055 & 1.29179 \
12 & 1.25403 & 1.28489 \
13 & 1.24837 & 1.27884 \
14 & 1.24339 & 1.27347 \
15 & 1.23895 & 1.26867 \
16 & 1.23498 & 1.26433 \
17 & 1.23138 & 1.26039 \
18 & 1.22810 & 1.25678 \
19 & 1.22510 & 1.25346 \
20 & 1.22233 & 1.25039
endarray
right)$$ For infinitely large values of $k$, the asymptotics of the estimate would be
$$x_est=1+frac1log left(kright)+gamma +1$$

For $k=1000$, the exact solution is $1.12955$ while the first approximation gives $1.11788$ and the second $1.11786$.

Using such estimates would make Newton method converging quite fast (shown below for $k=1000$).

$$left(
beginarraycc
n & x_n \
0 & 1.117855442 \
1 & 1.129429575 \
2 & 1.129545489 \
3 & 1.129545500
endarray
right)$$

Edit

We can obtain much better approximations if, instead of using a Taylor expansion of $g(x)$ to $Oleft((x-1)^2right)$, we build the simplest $[1,1]$ Padé approximant (which is equivalent to an $Oleft((x-1)^3right)$ Taylor expansion). This would lead to
$$x=1+ frac6 (k+k (psi ^(0)(k)+gamma )-1)pi ^2 k+6 (k+gamma (gamma
k+k-2)-1)-6 k psi ^(1)(k)+6 psi ^(0)(k) (2 gamma k+k+k psi
^(0)(k)-2)$$ Repeating the same calculations as above, the results are
$$left(
beginarrayccc
k & x_est & x_sol \
2 & 1.60000 & 1.58579 \
3 & 1.46429 & 1.46791 \
4 & 1.40435 & 1.41082 \
5 & 1.36900 & 1.37605 \
6 & 1.34504 & 1.35209 \
7 & 1.32741 & 1.33430 \
8 & 1.31371 & 1.32040 \
9 & 1.30266 & 1.30914 \
10 & 1.29348 & 1.29976 \
11 & 1.28569 & 1.29179 \
12 & 1.27897 & 1.28489 \
13 & 1.27308 & 1.27884 \
14 & 1.26787 & 1.27347 \
15 & 1.26320 & 1.26867 \
16 & 1.25899 & 1.26433 \
17 & 1.25516 & 1.26039 \
18 & 1.25166 & 1.25678 \
19 & 1.24844 & 1.25346 \
20 & 1.24547 & 1.25039
endarray
right)$$

For $k=1000$, this would give as an estimate $1.12829$ for an exact value of $1.12955$.

For infinitely large values of $k$, the asymptotics of the estimate would be
$$x_est=1+frac6 (log (k)+gamma +1)6 log (k) (log (k)+2 gamma +1)+pi ^2+6 gamma
(1+gamma )+6$$