TITLE: Few calculus questions
QUESTION [0 upvotes]: $\lim \dfrac{(-1+i\sqrt{3})^{3n+1}}{8^n-1}$

I have no idea how to find limits of complex numbers. 

$\sum \dfrac{(n!)^2}{2^{n^{2}}}x^n, x \in$ C 

I have no idea how to find limits of complex numbers. 

$\lim (\sqrt[3]{x^3+x^2+x+1}-\sqrt{x^2+x+1}*\dfrac{\ln(e^x+x)}{x})$

I tried multiplying to and bottom by $1/x$. But that yields $(1-1*1)/0$, which isn't useful. And l'Hospital's rule looks too long.
For these three, I have no idea at all...

$\sum ($arccot $n-\arcsin (1/n))$
$\sum (\cot (\dfrac{\pi n}{4n-2})-\sin \dfrac{\pi n}{2n+1})$
$\sum (\arccos (1-1/n) n^a), a \in $ R

Thanks! All sums and limits are to infinity. For sums, convergence/divergence is enough.
For the last three sums, WolframAlpha says that by the comparison test they converge, diverge and depending on $a$, respectively. However, I don't know what to compare them to.

REPLY [0 votes]: For the fourth question, expand  $ln(e^x+x)$ as a Taylor series at $x=0$. This gives $2 x-\frac{3 x^2}{2}$. Now, extract $x$ from each radical and remember that, if $y$ is small, $(1+y)^n$ can be approximated by $(1+n \space y)$. So, you should arrive to $-1$ but I let you doing something.
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