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Q:
Prove that $1 + \sqrt{2} + \sqrt{3} + \sqrt[4]{2} + \cdots \sqrt[n]{2} + \cdots
eq 2$.
Prove that $$1 + \sqrt{2} + \sqrt[4]{2} + \cdots \sqrt[n]{2} + \cdots
eq 2.$$
I don’t know what to do. My idea is to consider a sequence $\{a_n\}$ which is assumed to be a solution for this problem. Then, we have
$$a_0 + a_1\sqrt{2} + a_2\sqrt[4]{2} + \cdots + a_n\sqrt[n]{2} = 2$$
I don’t know how to proceed and would be grateful if you could point me in the right direction.
A:
I’m pretty sure that they are attempting to define $\lim\limits_{n\to\infty} a_n$ using the Tower Law, which states that if $r\gt0$ then $\sum\limits_{i=0}^n a_ir^i=a_n\frac{r^n}{r-1}$.
However, this is more or less what you have written as the first term of your expression.
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