<s>[INST] Ramanujan's radical and how we define an infinite nested radical I know it is true that we have
$$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}=3$$
The argument is to break the nested radical into something  like
$$3 = \sqrt{9}=\sqrt{1+2\sqrt{16}}=\sqrt{1+2\sqrt{1+3\sqrt{25}}}=...=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$
However, I am not convinced. I can do something like
$$4 = \sqrt{16}=\sqrt{1+2\sqrt{56.25}}=\sqrt{1+2\sqrt{1+3\sqrt{\frac{48841}{144}}}}=...=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$
Something must be wrong and the reason behind should be a misunderstanding of how we define infinite nested radical in the form of 
$$ \sqrt{a_{0}+a_{1}\sqrt{a_{2}+a_{3}\sqrt{a_{4}+a_{5}\sqrt{a_{6}+\cdots}}}} $$
I researched for a while but all I could find was computation tricks but not a strict definition. Really need help here. Thanks.
 [/INST] As others have said, the rigorous definition of an infinite expression comes from the limit of a sequence of finite terms. The terms need to be well-defined, but in practice, we just try to make sure the pattern is clear from context.
Now let's see what goes wrong with your other example. You wrote:
$$4 = \sqrt{16}=\sqrt{1+2\sqrt{56.25}}=\sqrt{1+2\sqrt{1+3\sqrt{\frac{48841}{144}}}}=...=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$
The problem is that each term in the sequence (e.g. if we stop at $4$) fails to include an "extra" amount (and this amount is not going to zero). So if we look at the partial sums, we see they won't converge to $4$ unless we include the extra amounts we keep pushing to the right. It's the same logical mistake as taking
\begin{align*}
  2 &= 1 + 1  \\
    &= \frac{1}{2} + \frac{1}{2} + 1  \\
    &= \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + 1 \\
    &= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + 1
\end{align*}
And then saying, wait, the partial sums in the last line only converge to $1$ instead of $2$. But this is a mistake because we can't push the extra $1$ "infinitely far" right. Otherwise, the partial sum terms we write down will just look like $\frac{1}{2} + \frac{1}{4} + \cdots$ and will never include the $1$. The same thing is happening (roughly) in your example.
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