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Determine the convergence or divergence of f(n) = (1 / n) e-πin / 2

Consider the function f(n) defined by

    \[ f(n) = \frac{1}{n} e^{-\frac{\pi i n}{2}}. \]

Determine the convergence or divergence of the sequence \{ f(n) \} and if it converges determine its limit.


This sequence converges since

    \begin{align*}  \lim_{n \to \infty} |f(n)| &= \lim_{n \to \infty} \left| \frac{1}{n} \cdot e^{-\frac{\pi i n}{2}} \right| \\[9pt]  &= \lim_{n \to \infty} \left( \frac{1}{n} \cdot \left| e^{-\frac{\pi i n}{2}} \right| \right). \end{align*}

But each of the limits in the product exists since

    \[ \lim_{n \to \infty} \frac{1}{n} = 0 \]

and

    \[ \lim_{n \to \infty} \left| e^{-\frac{\pi i n}{2}} \right| = \lim_{n \to \infty} 1  = 1. \]

Hence,

    \[ \lim_{n \to \infty} |f(n)| = 0 \cdot 1 = 0. \]

Therefore, \lim_{n \to \infty} f(n) = 0 (as we saw in this exercise, if f(n) is complex valued and \lim_{n \to \infty} |f(n)| = 0 then the sequence \lim_{n \to \infty} f(n) = 0 as well). Hence, the sequence \{ f(n) \} converges to 0.

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