Let \((s_n)_{n = 0}^{\infty}\) be a sequence of the form \(s_0 = \sqrt{a}, s_1 = \sqrt{a + \sqrt{a}}, s_2 = \sqrt{a + \sqrt{a + \sqrt{a}}}, ...\)
As n approaches infinity, the sequence approaches a "continued square root", a number of the following form:
x=a+a+a+...
I'll leave it as an exercise to show that if a>0, then this continued square root always converges, and the result is the positive root of the equation a=x2−x.
Let's take a=2 as an example. By the formula I just mentioned, we have 2+2+2+...=2.
One question I asked myself is, how fast do these continued square roots converge to 2?
To put it more precisely, let's consider the sequence 2−2,2−2+2,2−2+2+2, and so on. The decimal approximations for these numbers are 0.5858, 0.1522, 0.03843, 0.0096305, etc. Do you notice the pattern? The numbers seem to be getting about 4 times smaller each time.
Let's make this more rigorous. In the original sequence, we have sn+1=2+sn. Therefore, n→∞lim2−sn2−sn+1=n→∞lim2−sn2−2+sn=n→∞lim(2−sn)(2+2+sn)(2−2+sn)(2+2+sn)=n→∞lim(2−sn)(2+2+sn)4−(2+sn)=n→∞lim2+2+sn1=41
So, the differences between the continued square roots and their limit (2) do, in fact, approach a geometric sequence with 41 as the common ratio. Hence, sn can be asymptotically approximated by a function sn≈2−4nb.
Now, the million dollar question: What is this constant b?
In other words, what is n→∞lim4n(2−sn)?
I won't tell you the answer in case you want to solve it on your own. If you succeed, I'd love to see how you solved it so feel free to share it in the comments. If you'd like to see my solution, you can find it here.
Now let's generalize the previous result for the real number a. Let's replace 2 with x2−x, so that our sequence looks like this: sn={x2−x,(x2−x)+(x2−x),(x2−x)+(x2−x)+x2−x...} You can show (using a similar proof to mine above) that n→∞limx−snx−sn+1=2x1
Hence, the generalized sequence can be approximated by the function sn≈x−(2x)nLx where Lx is the following limit: Lx=n→∞lim(2x)n(x−sn)
Now, I have a conjecture for you. I haven't been able to prove/disprove it yet; see if you can:
Let sn be a sequence defined recursively by s0=x2−x and sn+1=(x2−x)+sn for n∈N, for some x>1. Let Lx=limn→∞(2x)n(x−sn). Then,
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@Ariel Gershon
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Let fn(y)=ntimesa+a+…+a+y, substitute y=a⋅g(x) to simplify fn(y). We can then replace a by t2−t and calculate Lt by letting n→∞. Then we can find the conjectured limit by letting t→∞. For example if we let a=2, k=1 we will get g(x)=2g2(2x)−1. One solution is g(x)=cos(x). From this we can calculate b=16π2.
@Ariel Gershon
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g(x) is an unknown function which satisfies g(x)=ag2(akx)−1. If we substitute y=a⋅g(x) in fn(y)=ntimesa+a+…+a+y, we have fn(y)=a⋅g(anknx). And if we replace a by t2−t, we can then calculate Lt as a function of t. This will simplify the conjectured limit Lt. If my reasoning is missing any steps, you can ask and I'll try to fix it.
@Ishan Singh
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OK, it makes more sense now as I was able to finish your argument. This is a very interesting approach! If you don't mind though, I'm going to simplify the equation for g to make it g(x)=ag2(cx)−1 (since k is an arbitrary constant, let c=ak).
There are two more stipulations for g and c which you didn't mention, but they are important. The proof by induction of this approach requires that the following be true:
(1) there must exist a real number z such that g(z)=aa
(2) g(cnz)≥0 for all n∈N
Also, we can write fn(a)=sn. Then by induction we can show that sn=a∗g(cnz). If we replace a with t2−t, then Lt will have the following formula: Lt=n→∞lim(2t)n(t−(t2−t)g(cnz))
In the case of t=2, we can have c=21 and g(x)=cos(x), since, as you mentioned, cos(x)=2cos2(2x)−1. We have z=4π since cos(4π)=22. Then we get L2=n→∞lim4n(2−2cos(4∗2nπ)) This limit does, in fact, evaluate to 16π2.
Interesting as this is, though, I think it won't be as easy to find g,c,z for t in general.
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Comments
Just to confirm, is b=16π2 ?
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Yes, very good!
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If we can solve the functional equation g(x)=ag2(akx)−1 where a is a fixed constant and k is arbitarary, we are almost done solving the conjecture.
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fn(y)=n timesa+a+…+a+y, substitute y=a⋅g(x) to simplify fn(y). We can then replace a by t2−t and calculate Lt by letting n→∞. Then we can find the conjectured limit by letting t→∞. For example if we let a=2, k=1 we will get g(x)=2g2(2x)−1. One solution is g(x)=cos(x). From this we can calculate b=16π2.
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g(x) and k?
I'm missing quite a few steps in your logic.... What areLog in to reply
g(x) is an unknown function which satisfies g(x)=ag2(akx)−1. If we substitute y=a⋅g(x) in fn(y)=n timesa+a+…+a+y, we have fn(y)=a⋅g(anknx). And if we replace a by t2−t, we can then calculate Lt as a function of t. This will simplify the conjectured limit Lt. If my reasoning is missing any steps, you can ask and I'll try to fix it.
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g to make it g(x)=ag2(cx)−1 (since k is an arbitrary constant, let c=ak).
OK, it makes more sense now as I was able to finish your argument. This is a very interesting approach! If you don't mind though, I'm going to simplify the equation forThere are two more stipulations for g and c which you didn't mention, but they are important. The proof by induction of this approach requires that the following be true:
(1) there must exist a real number z such that g(z)=aa
(2) g(cnz)≥0 for all n∈N
Also, we can write fn(a)=sn. Then by induction we can show that sn=a∗g(cnz). If we replace a with t2−t, then Lt will have the following formula: Lt=n→∞lim(2t)n(t−(t2−t)g(cnz))
In the case of t=2, we can have c=21 and g(x)=cos(x), since, as you mentioned, cos(x)=2cos2(2x)−1. We have z=4π since cos(4π)=22. Then we get L2=n→∞lim4n(2−2cos(4∗2nπ)) This limit does, in fact, evaluate to 16π2.
Interesting as this is, though, I think it won't be as easy to find g,c,z for t in general.
I want to point out that it only applies for a>0. (For a=0, it converges to the lower root of a=x2−x, namely x=0 instead of x=1.)
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Fair point. I'll fix it