For z_n=dfrac1+i^n2 : 1^° plot several terms; 2^° find its…

For $z_n=\dfrac{1+i^n}2$: $\ 1^\circ$ plot several terms; $\ 2^\circ$ find its accumulation points; $\ 3^\circ$ is the sequence bounded; $\ 4^\circ$ accumulation points of $|z_n|$; $\ 5^\circ$ find $\varlimsup|z_n|$ and $\varliminf|z_n|$.

Part $1^\circ$: the four repeating values

$$ z_1=\frac{1+i}2=\frac12+\frac12i, \quad z_2=\frac{1-1}2=0, \quad z_3=\frac{1-i}2=\frac12-\frac12i, \quad z_4=\frac{1+1}2=1 $$

and $z_{n+4}=z_n$ for all $n$.

Part $2^\circ$: accumulation points

Each of the four values recurs infinitely often (every $4$th term), so all four are accumulation points:

$$ \boxed{\left\{0,\ \frac12+\frac12i,\ \frac12-\frac12i,\ 1\right\}} $$

Part $3^\circ$: boundedness

$$ |z_1|=|z_3|=\frac{\sqrt2}2\approx0.7071, \quad |z_2|=0, \quad |z_4|=1 $$

All four values satisfy $|z_n|\le1$, so the sequence is bounded: $\boxed{|z_n|\le1 \text{ for all } n}$.

Part $4^\circ$: accumulation points of $(|z_n|)$

The modulus sequence cycles through the three distinct values $0,\ \frac{\sqrt2}2,\ 1$ (with $\frac{\sqrt2}2$ occurring twice per cycle, at $z_1$ and $z_3$), each recurring infinitely often:

$$ \boxed{\left\{0,\ \frac{\sqrt2}2,\ 1\right\}} $$

Part $5^\circ$: limsup and liminf

Since the tail $\{|z_k|:k\ge n\}$ always contains all three values $0,\frac{\sqrt2}2,1$ regardless of $n$ (the cycle keeps repeating forever):

$$ \boxed{\varlimsup_{n\to\infty}|z_n|=1, \qquad \varliminf_{n\to\infty}|z_n|=0} $$