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Determine the convergence or divergence of f(n) = (1 + i/2)-n

by RoRi

Consider the function f(n) defined by

    \[ f(n) = \left( 1 + \frac{i}{2} \right)^{-n}. \]

Determine whether the sequence \{ f(n) \} converges or diverges, and if it converges find the limit.

First, we note that

    \[ \left| 1 + \frac{1}{2}i \right| = \sqrt{1^2 + \left( \frac{1}{2} \right)^2} = \frac{\sqrt{5}}{2}. \]

Then,

    \begin{align*} \left| \left( 1 + \frac{i}{2} \right)^{-n} \right| &= \left| \frac{1}{\left(1 + \frac{i}{2} \right)^n} \right| \\[9pt] &= \left| \left( \frac{1}{1+\frac{i}{2}} \right)^n \right|\\[9pt] &= \left| \frac{1}{1+\frac{i}{2}} \right|^n \\[9pt] &= \left(\frac{2}{\sqrt{5}}\right)^n. \end{align*}

Therefore (since \frac{2}{\sqrt{5}} < 1 and is real) we know from (10.10) on page 380 of Apostol that

    \[ \lim_{n \to \infty} |f(n)| = \lim_{n \to \infty} \left( \frac{2}{\sqrt{5}} \right)^n = 0. \]

Hence, there exists an N such that for all n > N we have

    \[ \lim_{n \to \infty} |f(n) - 0| < \varepsilon \qquad \text{for all } \varepsilon> 0. \]

So \lim_{n \to \infty} f(n) = 0, so the sequence \{ f_n \} converges to 0.

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