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?

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TopNewestI 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 agoLog in to reply

– Steven Jim · 1 month ago

Thanks for fixing. I've been thinking hard ;)Log in to reply

– Swagat Panda · 1 month ago

Sorry for the inconvenience caused due to it. It was an absolute brainfade.Log in to reply

– Steven Jim · 1 month ago

Eh, it's all okay. Don't blame yourself.Log in to reply

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

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– Steven Jim · 1 month ago

I don't really understand what you wanted to mention. Can you please explain clearer?Log in to reply

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– Steven Jim · 1 month ago

I haven't mentioned that \(\sqrt{i}=-1\).Log in to reply

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– Swagat Panda · 1 month ago

But Steven never gave any solution, he was asking for one. He never said \( \sqrt{i}= -1\).Log in to reply

– Steven Jim · 1 month ago

What solution? I only asked a question!Log in to reply

– Elethelectric Penguin · 1 month ago

I'm terribly sorry, my browser has done something strange and my original reply was meant to be posted elsewhere. Please ignore it :(Log in to reply

– Steven Jim · 1 month ago

Eh, don't be stressed. We all have bad times XDLog in to reply

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

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– Steven Jim · 1 month ago

Can you please give reasons for your opinion?Log in to reply