Limit value of ${i^∞}$

Question

Can I investigate this limit and if yes, how? $${i^∞}$$

I am at a loss of ideas and maybe it is undefined?

Answer 1

Since the sequence $(i^n)_{n\in\mathbb N}$ is the sequence$$i,-1,-i,1,i,-1,-i,1,\ldots$$your sequence diverges.

Answer 2

You can't. The powers of $i$ form a periodic progression among four values. Convergence means you have some complex number for which the following is true. For any neighborhood of the number, the sequence eventually enters that neighborhood and never re-exits.

Answer 3

The convergence of certain sequences is impossible to find if the range is limited to a cycle of a few values. In this case $i^n$ is equal to either $\pm1$ or $\pm i$ with no other values. $\infty$ being the concept that it is makes it impossible to determine which of the four solutions would be $\lim_{n\to\infty}{i^n}$.

Other such expressions are $(-1)^n$ which cycles around $1$ and $-1$ in a similar fashion, or $\sin(x)$ and $\cos(x)$, whose range is limited to between $-1$ and $1$, without any convergence.

Answer 4

The limit doesn't exist in any event, but you should still be specific about what precisely is tending to $\infty$, because you could talk about the limit set (i.e., in this case, the set of subsequential limits), and then it matters. If the exponents are integers, the limit set is $\{1,-1,i,-i\}$. If the exponents are real numbers, the limit set is the unit circle. If the exponents are complex numbers (tending to infinity in absolute value) then the limit set is the complex plane.