5k views

### Limit of the nested radical $\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$ [duplicate]

Possible Duplicate: On the sequence $x_{n+1} = \sqrt{c+x_n}$ Where does this sequence converge? $\sqrt{7},\sqrt{7+\sqrt{7}},\sqrt{7+\sqrt{7+\sqrt{7}}}$,...
21k 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 ...
10k views

5k views

Give a precise meaning to evaluate the following: $$\large{\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dotsb}}}}}$$ Since I think it has a recursive structure (does it?), I reduce the equation to $$p=\sqrt{... 2answers 11k views ### calculate the limit of this sequence \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1..}}}} [duplicate] Possible Duplicate: \sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}, or the limit of the sequence x_{n+1} = \sqrt{c+x_n} i am trying to calculate the limit of a_n:=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+}}}}.. ... 2answers 12k views ### Convergence and limit of a recursive sequence x_{n+1}=\sqrt{p+x_n} [duplicate] Let p>0 and suppose that the sequence \{x_n\} is defined recursive as$$ x_1 = \sqrt{p}, \quad x_{n+1} = \sqrt{p + x_n}, $$for all n \in \mathbb{N}. How can I show that x_n converges, ... 3answers 1k views ### To prove a sequence is Cauchy [duplicate] I have a sequence:  a_{n}=\sqrt{3+ \sqrt{3 + ... \sqrt { 3} } }  , it repeats n-times. and i have to prove that it is a Cauchy's sequence. So i did this: As one theorem says that every ... 4answers 725 views ### Discuss the convergence of the sequence: a_1=1,a_{n+1}=\sqrt{2+a_n} \quad \forall n \in \mathbb{N} [duplicate] Computing the first few terms$$a_1=1, a_2=\sqrt{3}=1.732....,a_3=1.9318....,a_4=1.9828...$$I feel that (a_n)_{n\in \mathbb{N}} is bounded above by 2, although I have no logical reasoning for this. ... 3answers 493 views ### Different ways of computing \sqrt{1+\sqrt{1+\sqrt{1+\cdots}}} [duplicate] Possible Duplicate: On the sequence x_{n+1} = \sqrt{c+x_n} I am wondering how many different solutions one can get to the following question: Calculate \sqrt{1+\sqrt{1+\sqrt{1+\cdots}}} ... 4answers 3k views ### Prove convergence of sequence defined recursively a_{n+1} = \sqrt{6+a_n} [duplicate] We have a sequence:$$a_1=\sqrt{6}a_{n+1} = \sqrt{6+a_n}$$The problem is to check convergence and then find the limit. We know that sequence converges when it is monotonic and bounded. With a ... 4answers 424 views ### Proving that the sequence a_1 = 4, a_{n+1} = \sqrt{a_{n} +20} converges and finding its limit [duplicate] Given a sequence \{a_n\}, with n\geq1, where$$a_{1}=4,$$and$$a_{n+1}=\sqrt{a_{n} +20}.$$Prove via induction that, for all n \geq 1,$$a_{n+1}>a_{n}.$$Apparent Convergence ... 2answers 880 views ### Find the value of \sqrt{6+\sqrt{6+\sqrt{6+\dots}}} [duplicate] Evaluate$$\sqrt{6+\sqrt{6+\sqrt{6+\dots}}}$$I need some help with this question because I have no idea what is going on and help would be greatly appreciate :) 3answers 345 views ### Find the value of : \lim_{n\rightarrow \infty}\sqrt{2+\sqrt{2+\sqrt{…+\sqrt{2}}}} [duplicate] This is a nice limit$$\lim_{n\rightarrow \infty}\underbrace{\sqrt{2+\sqrt{2+\sqrt{...+\sqrt{2}}}}}_{n\text{ times}}$$and it is solved with well-known trigonometry formulas. The result is 2. The ... 2answers 252 views ### 'Continued roots'' [duplicate] Possible Duplicate: \sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}, or the limit of the sequence x_{n+1} = \sqrt{c+x_n} Some time ago I was playing with a calculator and I found the following relation$$2 ...

15 30 50 per page