How to plot the sum of this function series?

Question

Let us consider the sum of a function series $$ \sum _{n=1}^{\infty } \frac{z^n}{\left(z^n+1\right)^n}.$$

This series absolutely converges if Abs[z] < 1 or Abs[z] > 1 or z==1 in view of the results of

DiscreteMaxLimit[(Abs[z]^n/Abs[(1 + z^n)]^n)^(1/n), n -> Infinity, 
Assumptions -> Abs[z] < 1]

Abs[z]

and

DiscreteMaxLimit[(Abs[z]^n/Abs[(1 + z^n)]^n)^(1/n), n -> Infinity, 
Assumptions -> Abs[z] > 1]

0

and the root test. Its sum presents two different analytic functions having the unit disk $\{z:|z| < 1\}$ and its exterior $\{z:|z| > 1\}$ as the natural domains of existence (see Encyclopedia of Mathematics for info) since the term $\frac{z^n}{\left(z^n+1\right)^n}$ has $n$ poles of order $n$ uniformly distributed on the unit circumference.

ComplexPlot3D[Sum[z^n/(1 + z^n)^n, {n, 1, 20}], {z, 2}]

illustrates it.

In 14.0 on Windows 10 I try to plot this function for real values of $x$ from $0$ to $2$.

Here are my not very successful attempts taking a lot of time.

Plot[Sum[x^n/(1 + x^n)^n, {n, 1, Infinity}], {x, 0, 2}, PlotRange -> All]

Plot[NSum[x^n/(1 + x^n)^n, {n, 1, Infinity}], {x, 0, 2}, 
PlotRange -> All, WorkingPrecision -> 25]

Replacing the function under consideration by its partial sums, I obtain

Plot[{Sum[x^n/(1 + x^n)^n, {n, 1, 100}], Sum[x^n/(1 + x^n)^n, {n, 1, 1000}], 
Sum[x^n/(1 + x^n)^n, {n, 1, 10000}]}, {x, 0, 2}, PlotRange -> All]

and

General::munfl: 0.0000408571^71 is too small to represent as a normalized machine number; precision may be lost.

The latest approach is not reliable (think of Sum[x^n,{n,1,Infinity}] and Sum[x^n,{n,1,1000}]). The hard piece is that

Plot[{Sum[x^n/(1 + x^n)^n, {n, 1, 100}],  Sum[x^n/(1 + x^n)^n, {n, 1, 1000}], 
 Sum[x^n/(1 + x^n)^n, {n, 1, 10000}], 
 58 - 163 x + 156 x^2 - 50 x^3, 
 Sum[x^n/(1 + x^n)^n, {n, 1, 20000}]]}, {x, 0.95, 1.05}, PlotRange -> All]

General::munfl: 0.950002^13812 is too small to represent as a normalized machine number; precision may be lost.

Plot[NSum[x^n/(1 + x^n)^n, {n, 1, Infinity}], {x, 0.95, 1.05}, 
PlotRange -> All, WorkingPrecision -> 15, MaxRecursion -> 4, PerformanceGoal -> "Quality"]

NumericalMath`NSequenceLimit::seqlim: The general form of the sequence could not be determined, and the result may be incorrect.

We see the last two plots differ for $x<1$. 58 - 163 x + 156 x^2 - 50 x^3 is nothing but the wrong result of Series[Sum[x^k/(1 + x^k)^k, {k, 1, Infinity}], {x, 1, 3}] // Normal: if this series were convergent , then there is an analytic continuation through 1.

So the question is: how to reliably plot the function under consideration on the real range $[0,2]$?