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Imaginary isn't real, right?

It might be obvious that \(2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { ... } } } } } } \) equals 4. So what about \(i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { ... } } } } } } \)? The answer might be \(-1\), but I'm not sure as \(i\) is not a real number. Can anyone help?

Note by Steven Jim
1 month ago

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I don't know if it's absolutely correct, but I am posting it.

If we write \(i \) as \( e^{i\pi/2} \), then the given series becomes:

\( \begin{align} & e^{i\pi/2} \sqrt{e^{i\pi/2}\sqrt{e^{i\pi/2}\sqrt{e^{i\pi/2}\sqrt{e^{i\pi/2}} \cdots}}} \\ &= e^{i\pi \left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8} \cdots \right)} \\ & =e^{i\pi \left( \frac{1/2}{1-1/2} \right)} \\ & =\boxed{e^{i\pi}=-1} \end{align}\)

Edit: Sorry for the initial error, I wrote \(i=e^{i\pi} \), which was absolutely incorrect. It has been corrected now to \(e^{i\pi/2} \) Swagat Panda · 1 month ago

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@Swagat Panda Thanks for fixing. I've been thinking hard ;) Steven Jim · 1 month ago

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@Steven Jim Sorry for the inconvenience caused due to it. It was an absolute brainfade. Swagat Panda · 1 month ago

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@Swagat Panda Eh, it's all okay. Don't blame yourself. Steven Jim · 1 month ago

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Consider that sqrt(-1) = i not sqrt(i) = -1 Elethelectric Penguin · 1 month ago

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@Elethelectric Penguin I don't really understand what you wanted to mention. Can you please explain clearer? Steven Jim · 1 month ago

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@Elethelectric Penguin I haven't mentioned that \(\sqrt{i}=-1\). Steven Jim · 1 month ago

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@Elethelectric Penguin But Steven never gave any solution, he was asking for one. He never said \( \sqrt{i}= -1\). Swagat Panda · 1 month ago

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@Elethelectric Penguin What solution? I only asked a question! Steven Jim · 1 month ago

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@Steven Jim I'm terribly sorry, my browser has done something strange and my original reply was meant to be posted elsewhere. Please ignore it :( Elethelectric Penguin · 1 month ago

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@Elethelectric Penguin Eh, don't be stressed. We all have bad times XD Steven Jim · 1 month ago

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No it can't be Biswajit Barik · 1 month ago

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@Biswajit Barik Can you please give reasons for your opinion? Steven Jim · 1 month ago

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