Linked Questions

67
votes
4answers
12k views

$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$

(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12) For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x > 0$. Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and ...
7
votes
2answers
22k views

Show that $\sqrt{2+\sqrt{2+\sqrt{2…}}}$ converges to 2 [duplicate]

Consider the sequence defined by $a_1 = \sqrt{2}$, $a_2 = \sqrt{2 + \sqrt{2}}$, so that in general, $a_n = \sqrt{2 + a_{n - 1}}$ for $n > 1$. I know 2 is an upper bound of this sequence (I proved ...
3
votes
1answer
302 views

Assuming convergence of the following series, find the value of $\sqrt{6+\sqrt{6+\sqrt{6+…}}}$ [duplicate]

Assuming convergence of the following series, find the value of $\sqrt{6+\sqrt{6+\sqrt{6+...}}}$ I was advised to proceed with this problem through substitution but that does not seem to help unless ...
-3
votes
1answer
215 views

Find the value of $\sqrt{2+\sqrt{2+\sqrt{2+\dots}} }$ [duplicate]

How to prove that $\sqrt{2+\sqrt{2+\sqrt{2+\dots}} }=2$
2
votes
1answer
70 views

Real Analysis Sequence limit [duplicate]

Consider the following sequence: $$ \sqrt{2},\sqrt{2+\sqrt{2}},\sqrt{2+\sqrt{2+\sqrt{2}}} \cdots $$ a) Prove by induction that all terms of the sequence are bounded above by two. b) Show that this ...
20
votes
5answers
2k views

Where did the negative answer come from?

The question is to evaluate $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$ $$x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+x$$ $$x^2-x-2=...
12
votes
1answer
2k views

Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$

so I stumbled upon this equation/formula, and I have no idea how to prove it. I don't know how should I approach it: $$ \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{\vphantom{\large A}2\,}\,}\,}\,}\ =\...
3
votes
1answer
486 views

How to find this limit and prove it rigorously: $\sqrt{2 + \sqrt{2+\sqrt{2+\sqrt2…)}}}$? [duplicate]

$\sqrt{2 + \sqrt{2+\sqrt{2+\sqrt2...)}}}$. Pretty classic question, I think - and the limit is equal to 2. But how do I prove this rigorously? An epsilon-delta proof wouldn't work, since I wouldn't ...
3
votes
2answers
456 views

Construct a set of real numbers whose limit points comprise the set of integers $\mathbb{Z}$

My thought process is the following: Let $S=\{ m + \frac{1}{n}| m \in \mathbb{Z},n \in N \}$. Then I need to show that the limit points of $S$ are indeed the integers and that these are the only ...
1
vote
3answers
261 views

Show that the sequence $\sqrt{2}, \ \sqrt{2+\sqrt{2}}, \ \sqrt{2+\sqrt{2+\sqrt{2+}}}…$ converges and find its limit. [duplicate]

Setting $a_n=\sqrt{2+\sqrt{2+\sqrt{2+}}}\ $ I get that $$a_{n+1}=\sqrt{2+a_n} \quad \quad a_1=\sqrt{2}.$$ Clearly all numbers in the sequence are positive and we see that $a_n<a_{n+1} \ \forall \ n$...
0
votes
1answer
44 views

How do I prove that this recursive sequence converges? [duplicate]

Let $a_1=\sqrt2$ and let $a_n=\sqrt{2+a_{n-1}}$ for $n \ge 2$. How do I prove that this sequence converges?