Speaking information theoretically, if the matrix elements are integers, wouldn't they _have_ to blow up in order to converge to a real number with supposedly infinite information? At 12:01 AM 7/23/2012, Bill Gosper wrote:

We're used to evaluating an infinite continued fraction as a limit of ratios of elements of a running product of 2x2 matrices, while the actual elements of the product blow up.

E.g., for 4 arctan(1)

In[1153]:= FoldList[Dot, {{0, 4}, {1, 0}}, Table[{{2*k - 1, k^2}, {1, 0}}, {k, 9}]]

Out[1153]= {{{0, 4}, {1, 0}}, {{4, 0}, {1, 1}}, {{12, 16}, {4, 4}}, {{76, 108}, {24, 36}}, {{640, 1216}, {204, 384}}, {{6976, 16000}, {2220, 5100}}, {{92736, 251136}, {29520, 79920}}, {{1456704, 4544064}, {463680, 1446480}}, {{26394624, 93229056}, {8401680, 29675520}}, {{541937664, 2137964544}, {172504080, 680536080}}}

In[1154]:= N[Divide @@ # & /@ Rest[%]]

Out[1154]= {{4., 0.}, {3., 4.}, {3.16667, 3.}, {3.13725, 3.16667}, {3.14234, 3.13725}, {3.14146, 3.14234}, {3.14161, 3.14146}, {3.14159, 3.14161}, {3.14159, 3.14159}}

But here's an infinite matrix product where the elements converge individually:

Product[MatrixForm[{{(3*a^2 - 3*a*j + j^2)/(j*(-1 - 3*a + b + j)), a^3/(j*(-1 - 3*a + b + j))}, {1/j, 1}}], {j, b, ∞}] == MatrixForm[{{"?", (a^3*HypergeometricPFQ[{1 + a, 1 + a, 1 + a}, {1 + b, 1 + b}, 1])/b^2}, {"?", HypergeometricPFQ[{a, a, a}, {b, b}, 1]}}]

(Product means matrix product, not Mathematica's product.)

The ?s converge, too, but I don't know to what.

Note that the determinant ≠1, but approaches 1 in a very particular way.

It's not obvious to me how to scale a matrix to make its running product converge elementwise.

I actually have an elementwise convergent 2x2 product for the general 3F2[1], but idiot Mathematica 8.04 takes (literally?) forever to convert a simple sum of Pochhammers to a 3F2.

--rwg