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

**Answer**

\begin{eqnarray*}

x= a\sqrt { a\sqrt { a\sqrt { a\sqrt { a\sqrt { a\sqrt { \cdots } } } } } } \\

x=a^{ 1+1/2+1/4+1/8+\cdots} \\

x=a^2

\end{eqnarray*}

So it would seem that

\begin{eqnarray*}

i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { \cdots } } } } } }=\color{red}{-1}.

\end{eqnarray*}

**Attribution***Source : Link , Question Author : Community , Answer Author : Donald Splutterwit*