Author Topic: Q2 TUT 0301  (Read 588 times)

Victor Ivrii

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Q2 TUT 0301
« on: October 05, 2018, 06:14:25 PM »
Determine whether the given infinite series converges or
diverges:
\begin{equation*}
\sum_{n=1}^\infty \Bigl(\frac{1+2i}{\sqrt{6}}\Bigr)^n.
\end{equation*}

Meng Wu

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Re: Q2 TUT 0301
« Reply #1 on: October 05, 2018, 06:39:52 PM »
By Ratio Test:
$$\begin{align}\sum_{n=1}^\infty \Biggl(\frac{1+2i}{\sqrt{6}}\Biggl)^n &= \lim_{n\to\infty}\Biggl|\frac{\Big(\frac{1+2i}{\sqrt{6}}\Big)^{n+1}}{\Big(\frac{1+2i}{\sqrt{6}}\Big)^{n}}\Biggr|\\&=\lim_{n\to\infty}\Biggl|\frac{1+2i}{\sqrt{6}}\Biggr|\\&=\Biggl|\frac{1+2i}{\sqrt{6}}\Biggr|\\&=\sqrt{\Big(\frac{1}{\sqrt{6}}\Big)^2+\Big(\frac{2}{\sqrt{6}}\Big)^2}\\&=\sqrt{\frac{5}{6}}<1\end{align}$$
Therefore,  the given infinite series converges.