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Hurkyl
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#5
Mar1-04, 10:46 PM
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This is an interesting sort of problem because there is a hidden ambiguity.

A lot of times, when we write a limiting process with an ellipsis, such as

[tex]1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots[/tex]

the notation specifies the starting point of the iteration.


With these infinitely nested radicals, however, the initial point is not specified!


To write these more properly, we would do something like

[tex]
\begin{array}{l}
s_{k+1} = \sqrt{n + s_k} \\
s = \lim_{k \rightarrow \infty} s_k
\end{array}
[/tex]

But the missing piece, as mentioned, is where do we start; what value do we give for [itex]s_0[/itex]? Depending on what we choose for [itex]s_0[/itex] (and also depending on n), the convergence of this limiting process can vary wildly.