Question

Find the limit of the (10)^1/2+(10+(10)^1/2)^1/2.....

Answer 1

\[\sqrt{10}+\sqrt{10+\sqrt{10}}+\sqrt{10+\sqrt{10+\sqrt{10}}}\] is that the sum you're asking about? If so, its limit is infinity since each of the terms is larger than sqrt(10)

Answer 2

In retrospect, maybe you meant something else.
If you meant \[\sqrt{10 +\sqrt{10 +\sqrt{10+...}}}\],
then that does have a limit.

Call it L.
Then you see that upon squaring L, you get 10 + the limit L again.
So this limit L satisfies L^2 = 10 + L.
Therefore, if the limit L exists,
it must be the positive root of that quadratic equation.

We can show that L does exist by showing that
the sequence of partial sums is always increasing (which is true),
and that the whole thing is bounded above (ie. can’t become infinite).
by induction on the number of 10’s.
Since it’s always increasing, but is bounded above, then
it must reach some finite limit.