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U (n) = a / u (n-1) + b / u (n-2) + c / u (n-3) ....
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u(n)= (u(n-1)/u(n-2))+(u(n-2)/u(n-1)) this sequence has à limit of 2. Datas are the first numerators.
square root of the sum of a, b, c ... Here exemple for square root of 22(a=2, b=13 and c=7)
1, 2, 5, 41, 4181, 1405670, 10076524678201314263, 6, 510, 1037085, 1255440826885, 35700707871080941779883, 116972246657829092823242558810084906770
first datas are denominators of U (n) = 2/u(n-1) + 13/u(n-2) + 7/u(n-3)
U(1)=1, U(2)=2, U(3)=3
lim(U(n)) for n-->infinity Is the square root of the sum (2+13+7)
squared U(n)-->22.
We can generalize:
Let U(n)= a/u(n-1) + b/u(n-2) + c/u(n-3) +d/u(n-4)+ .....
a,b,c,d ... positive reals.
Lim(U(n)) for n --> infinity Is square root of the sum (a+b+c+d+...)
It is not necessary to order a, b, c .... There may be equality between a, b, or (and) c ...
Sequence may have only 2 terms:U(n)= a/u(n-1) + b/u(n-2).
Convergence is slow
u(n)= (a/u(n-1)+b/u(n-2))+(u(n-2)/uc(n-1)3)+ ....
~1, 2, 2.5, 2.05, 2.039512195122, 2.0000263080725, 2.0003822273707, 2.00000000316631
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first datas are denominators numerators of U (n) = 2/u(n-1) + 13/u(n-2) + 7/u(n-3)
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