Answer
strictly increasing
Work Step by Step
Given $$\left\{ \frac{2^n}{2^n+1}\right\}_{1}^{\infty}$$
Since $a_n=\dfrac{2^n}{2^n+1},\ \ a_{n+1}= \dfrac{2^{n+1}}{2^{n+1}+1}$, then
\begin{align*}
\frac{a_{n+1}}{a_n}&=\dfrac{2^{n+1}}{2^{n+1}+1}\frac{2^n+1}{2^n}\\
&= \dfrac{2(2^n+1)}{2^{n+1}+1}\\
&=\dfrac{(2^{n+1}+1)+1}{2^{n+1}+1}\\
&=1+\frac{1}{2^{n+1}+1}>1
\end{align*}
so, the sequence is eventually strictly increasing