Using College to Decipher a MATHCOUNTS Classic
by djmathman, Mar 8, 2016, 5:34 PM
It's always fun when you can take the things you learn in college math class and apply them in unexpected situations 
Quote:
Let 
be a fixed positive real number. Prove that ![\[\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}\]](//latex.artofproblemsolving.com/5/7/b/57b378c77e648c8154c3c13e99aa31f5a6b3df7d.png)
has a defined value. In other words, show that the infinite nested radical doesn't diverge.
be a fixed positive real number. Prove that ![\[\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}\]](http://latex.artofproblemsolving.com/5/7/b/57b378c77e648c8154c3c13e99aa31f5a6b3df7d.png)
has a defined value. In other words, show that the infinite nested radical doesn't diverge.
be a sequence of real numbers satisfying 
and 
for all 
.
goes to infinity becomes the expression we wish to compute (e.g. 
,
,
converge to some real value as 
.
be a sequence of real numbers which is monotonic (meaning it's either always increasing or always decreasing) and bounded. Then 
is non-decreasing; the case where the sequence is non-increasing is similar. Let 
denote the set of all 
)
,
that 
;
as 
.
such that for all 
we have 
.
for all 
implies 
for all 
is an upper bound for the set 
for all 
.
,
for some ![\[\sqrt{x+y_{k+1}}>\sqrt{x+y_k}\quad\implies y_{k+2}>y_{k+1}.\]](http://latex.artofproblemsolving.com/9/0/b/90b6422741ed4543c3e6cf21a00bf609d7e87d94.png)
Hence by induction we're done.
such that 
for all ![\[y_k<L=\frac{1+\sqrt{1+4x}}2,\]](http://latex.artofproblemsolving.com/b/d/d/bdd1708ea12011eb3e185556417e9689dd916422.png)
i.e. the solution in 
to the equation 
.
for some ![\[y_{k+1}=\sqrt{x+y_k}<\sqrt{x+L}=L,\]](http://latex.artofproblemsolving.com/1/d/9/1d981a999f9550c6298f060f97c1d2d64479d49d.png)
so by induction we're done.
This post has been edited 4 times. Last edited by djmathman, Mar 8, 2016, 6:07 PM
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