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Assume that k is positive (else the root of k will be a complex number), and the root is the positive square root.
Suppose this thing converges and the value it converges to is x. Then,
x^2 - k = x
So, x^2 - x - k = 0.
Solving, this has two solutions and one is negative. It can't be the negative one because everything is always positive by our assumption, so if x exists it must equal the positive root, i.e. x = (1+√(4k+1))/2.
It remains to show that the finite surds :√k,√(k+√k),... actually _do_ converge. Denote the n'th term by x_n. Observe x_n is increasing, so if we can show x_n is bounded above, we deduce convergence and the limit must be the x above.
Why is x_n bounded? Observe
(x_n)^2 - (x_n) < k (write out the left hand side)
Therefore, x_n < x because at x we have equality, x^2 - x = k.
So we're done!
If you drop the assumption and allow negative square roots and complex numbers, it's all going to depend on which of the two square roots you choose so it's not going to be well defined.
What else can I help you with?
What is an infinite surd?
Infinite surd is a term used in mathematics. The definition of an infinite surd is a never ending irrational number with an exact value that would be left in square root form.
How do you simplify a surd in the form A square root B?
A surd in the form a√b cannot, in general, be simplified.
How do you solve an infinite surd?
If an infinite surd is like (√(6+√(6+√(6+√(6...),to solve it, follow these steps: Set x = √(6+√(6+√(6+√(6... therefore x2 = 6+√(6+√(6+√(6+√(6... therefore x2 = 6+x therefore 6+x-x2 = 0. Factorising the expression gives you (-x+3)(x+2)=0 Only the positive answer need be concerned with: -x+3=0 therefore -x = -3 therefore x = 3.
What is a surd or irrational roots?
Irrational rootsRoots that are irrational are called surds. Irrational numbers are decimals that neither repeat nor terminate. But not all roots are surds. Sqrt(4) is not a surd, because sqrt(4) is + or - 2, which is rational. On the other hand, sqrt(2) is a surd, and that's because the square root of two is irrational.A surd is a number that cannot be changed into a fraction. They go on infinitely without any pattern.
What are surds?
A surd is a square root which cannot be reduced to a whole number: however you need to be able to simplify them.
What is an infinite surd?
Infinite surd is a term used in mathematics. The definition of an infinite surd is a never ending irrational number with an exact value that would be left in square root form.
What is the infinite surd of 2?
2
How do you find an infinite surd?
the same way you find chuck Norris
How do you simplify a surd in the form A square root B?
A surd in the form a√b cannot, in general, be simplified.
Is 5 a surd?
Yes it is a surd
What is a square root of surd?
If the value of the surd is positive, then it will be another surd. Otherwise it will be a complex number.
Cube root of 7 is a surd but is cube root of negative seven a surd?
I am pretty sure that the cube root of negative seven is a surd. I checked on the calculator.......and it showed a negative number.....??I think when it is not a surd...it's supposed to say error, so the number probably means that it is a surd..
Is there a law that you divide the surd index of a surd with a real no?
No. It is not a criminal or civil offence!
Who invented surd's?
i did
How do you convert Infinite numbers to fractions?
In general, you cannot. An infinite number divided by any non-zero number is still infinite. An infinite number divided by another infinite number may or may not be infinite.
What is area of plane in math?
In general, the plane is infinite in length and breadth and so infinite in area.
What does surd mean in maths?
A surd is an irrational number that can't be expressed as a fraction as for example the square root of 2
