Questions tagged [nested-radicals]
In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression.
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Finding the value of $\sqrt{z \sqrt{z \sqrt{z}}}…$
I was working on the following nested square root problem:
Let $a \in \mathbb R ^+$, what is the value of: $$\sqrt{a \sqrt{a \sqrt{a}}}...$$
I concluded that the answer is $a$ and then I thought ...
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$\sqrt{n+\sqrt{n+\sqrt{n+…+\sqrt{n+f(n)}}}}$
Let $\sqrt{n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}}=x$. Therefore, $$n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}=x^2$$ $$x^2-x-n=0$$ $$x=\sqrt{n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}}=\frac{1 \pm \sqrt{1+4n}}{2}...
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How to find the partial derivatives of the following nested expression?
I want to find the partial derivatives of the expression for $v_3(\boldsymbol{u})$ with respect to $u_1$, $u_2$ and $u_3$ from the expressions below. Here $\Phi$ denotes the cumulative distribution ...
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$\pi$, from Pentadecagon - infinitely expanding Balloon nested Radical
In this post, I would like to share the findings on derivation of $\pi$ with Pentadecagon inscribed in unit circle.
Here the side of each chord is $2\sin12^\circ$
(Bisecting the chord by segment which ...
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First few Prime numbers and nested radicals in association with $\pi$
There is interesting association between $2\cos(\frac{\pi}{60})$ which is $\frac{1}{2}\sqrt{8+\sqrt{15}+\sqrt3+\sqrt{10-2\sqrt5}}$ and first few Prime numbers with infinite expansion of balloon nested ...
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Simplifying $\sqrt{34+15\sqrt2}$ [closed]
$$\sqrt{34+15\sqrt2}$$
If we want $34+15\sqrt2$ to be a nice square $(a+b)^2=a^2+2ab+b^2$, most likely it is the case that $15\sqrt2$ corresponds to $2ab$. I don't know what to do from here. Is there ...
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Continued radical of powers of 4 equals 3 [closed]
Can someone explain to me why
$$\sqrt{4 + \sqrt{4^2 + \sqrt{4^3 + \sqrt{4^4 + \dots}}}} = 3???$$
I need an answer
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2answers
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$\pi$ & $\phi$ (Golden ratio), Pentagon inscribed in unit circle
Everyone is aware that square inscribed in unit circle and infinite product giving rise to $\pi$.
One of the simplest way to represent $\pi$ with the help of nested radical as follows
$$\pi = \lim_{n\...
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Nested radicals, Golden ratio, number series
Finite and infinite expansion of nested Radical involving Golden ratio as follows
$2\cos\frac{2\pi}{5} = \frac{1}{\phi} $ where $\phi$ is Golden ratio $(2\cos72Ā°)$
Let us expand this with increasing ...
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Interesting cyclic infinite nested square roots of 2 and cosine values
It is interesting to note that any angle between 45Ā° to 90Ā° satisfying $1\over4$ < $p \over q$ <$1\over2$ where $ p \over q$ is of form $p = 2^n $ and $q$ is an odd number satisfying $2^{n+1} &...
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Infinite Nested Radical
Ramanujan discovered that $$x+n+a=\sqrt{ax+(n+a)^2+x\sqrt{a(x+n)+(n+a)^2+...}}$$ (see equation (27) here). I didn't understand how we can use this (basically what to put in place of $x, n, a$) to ...
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1answer
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How to calculate $x_{n}= \sqrt{n+ \sqrt{n- 1+ \sqrt{n- 2+ \sqrt{\cdots + \sqrt{2+ \sqrt{1}}}}}}$ without using surd signs
How to calculate:
$$x_{n}= \sqrt{n+ \sqrt{n- 1+ \sqrt{n- 2+ \sqrt{\cdots + \sqrt{2+ \sqrt{1}}}}}}$$
without using surd signs.
My attempt:
I saw that
$$x^{2}_{n}= n+ x_{n- 1}$$
Therefore
$$x= 1+ \frac{...
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Is $\sqrt[3]{7+5 \sqrt{2}}+\sqrt[3]{20-14 \sqrt{2}}$ a rational number?
Is there a way to show that
$$\alpha=\sqrt[3]{7+5 \sqrt{2}}+\sqrt[3]{20-14 \sqrt{2}}$$
is a rational number?
I found $\alpha=3$ from doing simplifications. But, I would like to known a different ...
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Differents ways to evaluate the sum $\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{12+\cdots}}}}$
Evaluate $$\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{12+\cdots}}}}$$
My approach:
Let $$x=\sqrt{12+ \sqrt{12+\sqrt{12+\cdots}}}$$
so, we have that $$\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{12+\cdots}}}}\iff \sqrt{12+...
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Peculiar Nested radicals, cosine values, Jacobsthal-Lucas numbers
Let us consider half angle formula for equilateral triangle in unit circle with initial angle as $\frac{\pi}{3}$
$2\cos\frac{\pi}{6} = \sqrt{2+2\cos\frac{\pi}{3}} = \sqrt{2+1}$
Further half angle when ...
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0answers
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Golden ratio, cyclic infinite nested square roots of 2 and and Geometric progression and cosine value
In connection to previous posts here and herenow we know $2\cos\frac{2\pi}{5}$ can be represented as $\sqrt{2-\sqrt{2+...}}$ simply represented as $cin\sqrt2[1-1+]$ and this converges to inverse of ...
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For a recursion based on $x+y+z+2\sqrt{xy+yz+zx},$ does what happens in $\mathbb Z[\sqrt n]$ stay in $\mathbb Z[\sqrt n]$?
This problem has a geometric origin which I'll outline below, but I believe the concepts and explanation are algebraic.
Given a function on triples
$$K((x,y,z))=x+y+z+2\sqrt{xy+yz+zx}$$
we build a ...
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0answers
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Cosine angles and finite nested square roots of 2
Let us consider interesting factors about cosine angles
$2\cos(\frac{\pi}{2^3}) = \sqrt{2+\sqrt2}$
$2\cos(\frac{\pi}{2^4}) = \sqrt{2+\sqrt{2+\sqrt2}}$
To generalizeĀ
$2\cos(\frac{\pi}{2^n}) = \sqrt{2+\...
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Interesting solutions for cyclic infinite nested square roots of 2
I have derived cosine values for following cyclic infinite nested square roots of 2 ( Hereafter simply referred as $cin\sqrt2$)
$cin\sqrt2[1-]$ represents $\sqrt{2-\sqrt{2-...}}$
$cin\sqrt2[1-1+]$ ...
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Solving finite and infinite nested square roots of 2 - yet another interesting approach
Consider the following consecutive equalities:
$\sqrt2=2\cos(\frac{1}{4})\pi$
$\sqrt{2-\sqrt2}=2\cos(\frac{3}{8})\pi$
$\sqrt{2-\sqrt{2-\sqrt{2}}}=2\cos(\frac{5}{16})\pi$
$\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{...
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1answer
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Interesting infinite nested square roots of 2 for $2\cos1°$ and $2\sin1°$
It is interesting to note that any angle between 45Ā° to 90Ā° satisfying $1\over4$ < $p \over q$ <$1\over2$ where $ p \over q$ is of form $p = 2^n $ and $q$ is an odd number satisfying $2^{n+1} &...
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Solving $\sqrt{2+\sqrt{2-\sqrt{2+x}}} = x$.
ETA: since commenters noted I missed a sign I am correcting that.
This is one of those Mathsolutionzz instagram problems. I was curious if I did this correctly, because I looked at the comments, and ...
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1answer
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Infinite ways to represent $\pi$ as product of nested square roots of $2$ and $2^n$ and odd numbers
Polygon inscribed in a circle leads to famous infinite product by Viete's formula.here
One way to represent that as infinite nested radical of 2 as follows
$$\pi = \lim_{n\to\infty}2^n \times \sqrt{2-\...
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Solve the inequality $\sqrt[3]{1+(3+x)\sqrt{x}+3x} - \sqrt[3]{1-(3+x)\sqrt{x}+3x} > x + a$
So am trying to solve this inequality, $\sqrt[3]{1+(3+x)\sqrt{x}+3x} - \sqrt[3]{1-(3+x)\sqrt{x}+3x} > x + a$, the problem is of course for values $0<a<1$, I tried working through it multiple ...
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2answers
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An arithmetic problem that the sum of two irrationals involving cube roots makes an integer
Consider the following expression:
$$(20-\sqrt{392})^{1/3}+(20+\sqrt{392})^{1/3}$$
This equals $4$, but how can I show this?
Note that I do not want to make use of the following line of reasoning: 4 ...
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1answer
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Find derivative of $f(x)=\tan^2x \sqrt{\tan x\sqrt[3]{\tan x\sqrt[4]{\tan x…}}}$
Find $f(x)$ for $f(x)=\tan^2x \sqrt{\tan x\sqrt[3]{\tan x\sqrt[4]{\tan x...}}}$
I found that the right expression is an infinite series of $\sum^{\infty}_{n=2}{\frac{1}{n!}}$. I know the series ...
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2answers
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How to prove $f(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}$ is differentiable?
I've been working on this question for over 2 hours now and I tried to use the limit definition of a derivative to show that it's differentiable but that got me nowhere because I was completely ...
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1answer
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How to denest $\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac19}-\sqrt[3]{\frac29}+\sqrt[3]{\frac49}$ from scratch?
I have seen several questions asking for the proof of
$$\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac19}-\sqrt[3]{\frac29}+\sqrt[3]{\frac49}$$
However, I want to simplify $\sqrt[3]{\sqrt[3]{2}-1}$ into the ...
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Nested radicals like Ramanujan's infinite radicals
$$\sqrt{1+\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+...}}}}}=?$$
This like the
My question is about "how to start this problem ?".I've been thinking for over two hours but get stuck at the end. I ...
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1answer
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Taylor series expansion and value of cosine angle
This is a related question in deriving cosine values by 2 different means
Deriving values of Trigonometric angles will be easier with Taylor series expansion for first few terms for some of the ...
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4answers
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Rationalize the denominator of $2\over{2 - \sqrt[4]{2}}$?
Rationalize the denominator of $2\over{2 - \sqrt[4]{2}}$.
Here's my progress. Let $x = \sqrt[4]{2}$. Then our expression can be written as $x^4/(x^4 - x)$, which simplifies to $x^3/(x^3 - 1)$. ...
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1answer
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Does the sequence $t_n=\sqrt{a_1+\sqrt{a_2+\cdots+\sqrt{a_n}}} (a_k>0,k=1,2,\cdots)$ converge? [duplicate]
Does the sequence $t_n=\sqrt{a_1+\sqrt{a_2+\cdots+\sqrt{a_n}}} (a_k>0,k=1,2,\cdots)$ converge?
I need to prove that if
$$ \limsup\limits_{n\to\infty} \dfrac{\ln\ln a_n}{n}<\ln 2, $$
the sequence ...
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2answers
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Prove that $\frac{a}{1+a^2}+\frac{b}{1+a^2+b^2}+\frac{c}{1+a^2+b^2+c^2}+\frac{d}{1+a^2+b^2+c^2+d^2}\leq\frac{3}{2}$
For any reals $a$, $b$, $c$ and $d$ prove that:
$$\frac{a}{1+a^2}+\frac{b}{1+a^2+b^2}+\frac{c}{1+a^2+b^2+c^2}+\frac{d}{1+a^2+b^2+c^2+d^2}\leq\frac{3}{2}$$
C-S in the IMO 2001 stile does not help here:...
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Infinite Nested Square Root, Prove or disprove that there is at least one real number 𝑥 satisfy
$$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}} = 4$$
If I let $X_1=\sqrt x$, $X_2= \sqrt {x+X_1}$, then $X_{n+1}= \sqrt{x+X_n}$
It is a increasing sequence since $X_n<X_{n+1}$ for all $n$.
However, is ...
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1answer
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Representing $\cos(\frac{Ļ}{11})$ as cyclic infinite nested square roots of $2$
How can we represent $2\cos(\frac{\pi}{11})$ and $2\cos(\frac{\pi}{13})$ as cyclic infinite nested square roots of 2
I have partially answered below for $2\cos(\frac{\pi}{11})$ which I derived it ...
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$\sqrt{a^2+5b^2}+\sqrt{b^2+5c^2}+\sqrt{c^2+5a^2}\geq\sqrt{10(a^2+b^2+c^2)+8(ab+ac+bc)}$ for any real numbers.
I think that this inequality is strong, though I do not have knowledge of many techniques. There goes my work:
Positive variables only make the inequality stronger, hence suppose $a,b,c\geqslant0$
$$
\...
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4answers
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Simplify $(1+\sqrt{3}) \cdot \sqrt{2-\sqrt{3}}$
Can someone help me simplify $(1+\sqrt{3})\times\sqrt{2-\sqrt{3}}$?
The end result is $\sqrt{2}$, however, I honestly do not know how to get there using my current skills.
I asked a teacher/tutor and ...
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1answer
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Solving cyclic infinite nested square roots of 2 as cosine functions
Common infinite nested square roots of 2 are well known from school grade.
We used to solve $$\sqrt{2+\sqrt{2+\sqrt{2+...}}}$$ as
$x=\sqrt{2+x}$ which becomes $x^2 = x+2$ ==> $x^2-x-2=0$
The ...
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1answer
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Prove the convergence of the infinite product involving nested radicals [duplicate]
How to show that $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2+\sqrt{2}}}{2}\cdot \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot ... =\prod\limits_{n\in \mathbb{N}} \frac{1}{2} ~ \overbrace{\sqrt{2+\sqrt{2+...+\...
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2answers
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Find value of $\sin x-\frac{1}{\cot x}$
If $\sin x+\frac{1}{\cot x}=3$, calculate the value of $\sin x-\frac{1}{\cot x}$
Please kindly help me
Let $\sin x -\frac{1}{\cot x}=t$
Then, $$\sin x= \frac{3+t}{2}, \cot x= \frac{2}{3-t}$$
By ...
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1answer
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Is $\sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}$ even? prime? composite? an even prime? [duplicate]
I am not being able to solve this question. It is basically a high school level question and chapter I think is basic fundamentals of maths.
The value of
$$\sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}...
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2answers
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Solving infinite nested square roots of 2 converging to finite nested radical
Can anyone explain to solve the identity posted by my friend $$2\cos12Ā°= \sqrt{2+{\sqrt{2+\sqrt{2-\sqrt{2-...}}} }}$$ which is an infinite nested square roots of 2. (Pattern $++--$ repeating ...
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1answer
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Alternate method to solve $\sqrt{11\sqrt{11\sqrt{11…4\, \text{times}}}}$
Question :
What is the value of
$$\sqrt{11\sqrt{11\sqrt{11...4\,\text{times}}}}$$
I did it by solving square root one by one.
$$\sqrt{11\sqrt{11\sqrt{11\times11^\frac{1}{2}}}}$$
$$\sqrt{11\sqrt{11\...
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3answers
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Does a value for $\sqrt{x+\sqrt{x+\sqrt{x+…}}}$ actually exist? [duplicate]
I have seen questions of this type being solved as follows :
$\sqrt{x+\sqrt{x+\sqrt{x+...}}}$'s value does not change if we add an $x$ to the expression and square root it. Let the value of this ...
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2answers
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Infinite nested radical with variables
Let $a \bowtie b = a+\sqrt{b+\sqrt{b+\sqrt{b+...}}}$. If $4\bowtie y = 10$, find the value of $y$.
I know that $\sqrt{y+\sqrt{y+\sqrt{y+...}}}=6,$ but what do I do know? I know how to solve normal ...
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1answer
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Simplify a radical inside a radical inside a radical
The question is this:
Simplify $\sqrt{1+\sqrt{21+12\sqrt{3}}}$
I defined the value as x, then got $(x+1)^2(x-1)^2$=$21+12\sqrt{13}$.
I don't know what to do from there. Any help?
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1answer
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How to get rid of such radicals?
I would like to know if there is any way I can get rid of these cubic radicals bellow (1). I am allowing both complex and real values.
$$ \sqrt[3]{ -\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27} }...
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2answers
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An analogue to Ramanujan's identity $\left\{3\left(\left(a^{3}+b^{3}\right)^{1/3}-a\right)\left(\left(a^{3}+b^{3}\right)^{1/3}-b\right)\right\}^{1/3}$
In this paper discussing Ramanujan's submitted questions to the Mathematical journal, Question 785 states:
Show that;
$$\small{\left\{3\left(\left(a^{3}+b^{3}\right)^{1/3}-a\right)\left(\left(a^{3}+b^...
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1answer
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How would I simplify this function $\rho(x)=x+\sqrt{x-\sqrt{x-\sqrt{x+\sqrt{\dots}}}}$
How do I simplify $\rho(x)$ into simple terms?
$$\rho(x)=x+\sqrt{x-\sqrt{x-\sqrt{x+\sqrt{x-\sqrt{x+\sqrt{x+\sqrt{x-\sqrt{\dots}}}}}}}}$$
where the subtracting and the adding follows the ThueāMorse ...
12
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3answers
283 views
Simplify the radical $\sqrt{x-\sqrt{x+\sqrt{x-…}}}$
I need help simplifying the radical $$y=\sqrt{x-\sqrt{x+\sqrt{x-...}}}$$
The above expression can be rewritten as $$y=\sqrt{x-\sqrt{x+y}}$$
Squaring on both sides, I get $$y^2=x-\sqrt{x+y}$$
...
