$1^{\infty}$

The number $1$ is a real number and it involves in multiplication in some cases. When the number $1$ is multiplied by itself, the product of them is equal to one.

$1$ $\times$ $1$ $\,=\,$ $1$

In this case, the number of factors is two. Hence, the product of ones can be written as the one raised to the power of two as per exponentiation.

$\therefore\,\,\,\,$ $1^2$ $\,=\,$ $1$ $\times$ $1$ $\,=\,$ $1$

Observe the following three examples

$(1).\,\,$ $1^3$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\,=\,$ $1$

$(2).\,\,$ $1^4$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\,=\,$ $1$

$(3).\,\,$ $1^5$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\,=\,$ $1$

The number $1$ can be multiplied by itself as many times as we want but their product is always equal to $1$ in all the cases.

Let us assume that the number $1$ is multiplied by itself infinite times.

$1^\infty$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $\cdots$ $\times$ $1$

According to the above examples, we can think that the one raised to the power of infinity is also equal to one. Logically, it can be true but it is incorrect practically.

The exponent is infinite in this case. It means, the number of times the number $1$ should be multiplied by itself is indeterminate. So, it is not possible to perform the multiplication infinite times.

$\therefore\,\,$ $1^\infty$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $\cdots$ $\times$ $1$ $\,\ne\,$ $1$

The value of $1$ raised to the power of infinite is indeterminate and the $1$ raised to the power of infinity is called an indeterminate form in mathematics.

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