\(\begin{align} A_1 & = \sqrt{3} \\ A_2 & = \sqrt{A_1} \\ A_3 & = \sqrt{A_2} \\ & \vdots \\ A_{n+1} &= \sqrt{A_n} \end{align}\)
What is the value of the infinite product \(A_1 \times A_2 \times A_3\times \cdots \, ?\)
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
+23245 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.
