Can You Help Prove These Mathematical Identities?

[guysensei1]

Basically, I was messing with my calculator,
and i thought of this:

sqrt(x^sqrt(x^sqrt(x^...)))=sqrt(x)^sqrt(x)^sqrt(x)^...

where they go infinitely.
I can't prove it, so I need help.

another thing is 1/(a)+1/(a^2)+1/(a^3)... = 1/(a-1)

Perhaps the proof is something really obvious I missed?

 

[bossman27]

I'm not sure about the first off the top of my head, but I'll help with the second:

What you have here is a geometric series: $1 + r + r^{2} + \cdots + r^{n}$ with $r = 1/a$

Notice that you don't have the 1, so denoting the sum as $S$, and since $(1 + r + r^{2} + \cdots + r^{n})(1-r) = 1 - r^{n+1}$

$S+1 = \lim_{n \rightarrow \infty} \frac{1 - r^{n+1}}{1-r}$

This reduces to:

$S+1 = \frac{a}{a-1} \Rightarrow S = \frac{a}{a-1} - \frac{a-1}{a-1} = \frac{1}{a-1}$

Edit: typo

 

[bossman27]

Actually I think the first one is deceptively simply. You know that $(x^{m})^{n}= x^{mn}$, so think of the square roots simply as $\frac{1}{2}$ exponents.

For a finite example: $[x^{x^{1/2}}]^{1/2} = x^{(1/2)x^{1/2}} = (x^{1/2})^{x^{1/2}}$

Of course, this logic carries for an infinite number of exponents as well.