Compute the infinite product
[sin(x) cos(x/2)]^1/2 · [sin(x/2) cos(x/4)]^1/4 · [sin(x/4) cos(x/8)]^1/8 · ... ,
where 0 ≤ x ≤ 2π
After modifying the program to use degree input converted to radian and finding that 90° resulted in a value of .5, my first thought was the formula gave sin(x/3), but that didn't pan out for other values. The next thought was sin(x)/2. That did work:
list
10 for Xn=1 to 20:X=Xn/10
20 Deno=1:Deno2=2:A=1
30 for F=1 to 50
40 Fctr=(sin(X/Deno)*cos(X/(Deno2)))^(1/(Deno2))
50 A=A*Fctr
70 Deno=Deno2:Deno2=Deno2*2
80 next
90 print X,using(1,8),A,asin(A),sin(X)/2
100 next
OK
x inf. product arcsin(prod) sin(x) / 2
0.1 0.04991671 0.04993746 0.04991671
0.2 0.09933467 0.09949876 0.09933467
0.3 0.14776010 0.14830313 0.14776010
0.4 0.19470917 0.19596094 0.19470917
0.5 0.23971277 0.24206998 0.23971277
0.6 0.28232124 0.28621292 0.28232124
0.7 0.32210884 0.32795621 0.32210884
0.8 0.35867805 0.36685132 0.35867805
0.9 0.39166345 0.40243879 0.39166345
1.0 0.42073549 0.43425591 0.42073549
1.1 0.44560368 0.46184848 0.44560368
1.2 0.46601954 0.48478658 0.46601954
1.3 0.48177909 0.50268383 0.48177909
1.4 0.49272486 0.51521835 0.49272486
1.5 0.49874749 0.52215311 0.49874749
1.6 0.49978680 0.52335261 0.49978680
1.7 0.49583241 0.51879310 0.49583241
1.8 0.48692382 0.50856438 0.48692382
1.9 0.47315004 0.49286296 0.47315004
2.0 0.45464871 0.47197777 0.45464871
OK
The infinite product column and the sin(x)/2 column match.
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Posted by Charlie
on 2013-12-29 14:56:59 |
computer solution
