Let $a>0$ .
How we can find the limit of :
$$\sqrt{a+\sqrt{2a+\sqrt{3a+\ldots}}}$$
Thanks in advance for your help
Answer
In general, this is not known (as far as I know, at least). This is an extension of the so-called Kasner Number, and there is a good paper on the problems of finding closed forms of the Number can be found here (the paper also discusses infinite nested radicals).
This sequence does converge, at least. It's monotone increasing and bounded, so that's handy.
As an interesting note, many people are familiar with the work of Kasner without knowing it. It was Kasner who gave the name "googol" to the number 1 followed by 100 zeroes. Cool.
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