Test $x_n = (n+i\pi)^n n^{-n + 1/n}$ for convergence and give its limit if possible.

Question

Test $x_n = (n+i\pi)^n n^{-n + 1/n}$ for convergence and give its limit if possible.

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I'm not really sure what to do here. My first instinct was to rewrite the sequence as $x_n= (n+i\pi)^n n^{-n} n^{1/n}$ and evaluate the limits, but I'm left with $\lim_{n\rightarrow\infty} n^{1/n}=1$ and $\lim_{n\rightarrow\infty} n^{-n}=0$, which leaves me with nothing really. Can somebody help out?

Answer 1

In my usual naive way,

$\begin{array}\\ x_n &= (n+i\pi)^n n^{-n + 1/n}\\ &= (1+i\pi/n)^n n^{1/n}\\ &\to e^{\pi i} \qquad\text{since }(1+x/n)^n \to e^x \text{ and } n^{1/n} \to 1\\ &=-1\\ \end{array} $