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HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
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"Beeler, M., Gosper, R.W., and Schroeppel, R. HAKMEM . MIT AI
Memo 239, Feb. 29, 1972.
Retyped and converted to html ('Web browser format) by
Henry Baker,
April, 1995.
CONTINUED FRACTIONS
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ITEM 97 (Schroeppel): CONTINUED FRACTIONS
Simple proofs that certain continued fractions are sqrt(2), sqrt(3),
etc. Proof for sqrt(2):
X = [1, 2, 2, 2, ...]
(X-1) (X+1) = [0, 2, 2, 2, ...] * [2, 2, 2, 2, ...] = 1
2
X - 1 = 1
X = sqrt(2)
Proof for sqrt(3):
Y = [1, 1, 2, 1, 2, ...]
(Y + 1) (Y - 1) = [2, 1, 2, 1, 2, ...] * [0, 1, 2, 1, 2, ...]
= 2 * [1, 2, 1, 2, 1, ...] * [0, 1, 2, 1, 2, ...] = 2
2
Y - 1 = 2
Y = sqrt(3)
Similar proofs exist for sqrt(5) and sqrt(6); but sqrt(7) is hairy.
ITEM 98 (Schroeppel):
The continued fraction expansion of the positive minimum of the
factorial function (about 0.46) is
[0, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 43, ...].
ITEM 99 (Schroeppel):
The value of a continued fraction with partial quotients increasing in
arithmetic progression is
I (2/D)
A/D
[A+D, A+2D, A+3D, ...] = -------------
I (2/D)
1+(A/D)
where the I 's are Bessel functions. A special case is
I (2)
0
[1, 2, 3, 4, ...] = ----- .
I (2)
1
ITEM 100 (Perron):
n
/===\
! ! 1
! ! (1 + --) =
! ! Ak
k = 1
1 (A1 + 1)A1 (A2 + 1)A2 (A(n-1) + 1)A(n-1)
1 + ----- ---------- ---------- ... ------------------ .
A1 - A1+A2+1 - A2+A3+1 - A(n-1)+An+1
ITEM 101A (Gosper):
On the theory that continued fractions are underused, probably because
of their unfamiliarity, I offer the following propaganda session on
the relative merits of continued fractions versus other numerical
representations. For a good cram course"
....
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