In[]:=
deploy
Thu 1 Jul 2021 09:42:56
https://math.stackexchange.com/questions/4180259/behavior-of-inverse-of-fs-frac1h-n-sum-i-1n-left1-frac1i-right
In[]:=
eps=
-2
10
;​​n=1000;​​solveExact[p_,n_,eps_]:=​​formula[s_]=Sum
s
1-
1
p
i
1
p
i
,{i,1,n}HarmonicNumber[n,p];​​invert[f_]:=FindRoot[f[s]eps,{s,2},WorkingPrecisionMachinePrecision+Log[n]];​​s/.invert[formula]​​;​​solveIntegralExact[p_,n_,eps_]:=​​f[s_]=
Gamma
-1+p
p
Gamma[1+s]
pGamma2-
1
p
+s
-
1-p
n
Hypergeometric2F1
-1+p
p
,-s,2-
1
p
,
-p
n

-1+p
HarmonicNumber[n,p];​​invert[f_]:=FindRoot[f[s]eps,{s,100},WorkingPrecision1000,MaxIterations300];​​SetPrecision[s/.invert[f],7]​​​​;​​{solveExact[101/100,n,eps],solveIntegralExact[101/100,n,eps]}
Out[]=
{1786.3188978697341942786,1785.420}