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\(\begin{align} A_1 & = \sqrt{3} \\ A_2 & = \sqrt{A_1} \\ A_3 & = \sqrt{A_2} \\ & \vdots \\ A_{n+1} &= \sqrt{A_n} \end{align}\)

What is the value of the infinite product \(A_1 \times A_2 \times A_3\times \cdots \, ?\)

 Jul 6, 2020
 #1
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s=1;a=1;listfor(i, 1, 25,s=(2#(3)) * 2#s

 

OUTPUT =(1.732050808, 2.279507057, 2.615056629, 2.800923042, 2.898753029, 2.948942029, 2.974361459, 2.987153223, 2.99356972, 2.996783135, 2.998391136, 2.99919546, 2.999597703, 2.999798845, 2.999899421, 2.99994971, 2.999974855, 2.999987427, 2.999993714, 2.999996857, 2.999998428, 2.999999214, 2.999999607, 2.999999804, 2.999999902) =It converges to 3

 Jul 6, 2020
 #2
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Let  A1·A2·A3· ...  =  X

 

Square both sides:

     ( A1·A2·A3· ... )2  =  X2

      (A1)2·[ (A2)2·(A3)2·(A4)2· ... ]  =  X2 

 

But:  (A1) = 3     (A2)2 = A1     (A3)2 = A2     (A4)2 = A3  

 

So:  (A1)2·[ (A2)2·(A3)2·(A4)2· ... ]  =  3·[ A1·A2·A3· ... ]  =  X2  

and:  A1·A2·A3· ...  =  X   --->         =  3·[ X ]  =  X2     --->   3  =  X

 

So, the product is 3.

 Jul 6, 2020

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