It seems that

Limit(N+(sum((-1)^n*Sum(1/n^x, x = 1 .. N), n = 1 .. infinity)), N = infinity)=log(2)

 evalf(300+sum((-1)^n*(Sum(1/n^x, x = 1 .. 300)), n = 1 .. infinity), 30)

gives

0.693147180559945309417232121.

 sum(1/n^x, x = 1 .. infinity)

gives

1/(n-1).

There seems to be patterns for sin(10^-k) for rational k;

Here we have the "floats."

n sin(10^(-n-1/2))

1 0.03161750640

2 0.003162272390

3 0.0003162277607

4 0.00003162277660

5 0.000003162277660

6 0.0000003162277660

7 0.00000003162277660

More later on using the mantissa. You're welcome to join me.

I define a partial repeating decimal as shown in the following example: if you have the decimal expansion 0.1728394877777777777777777777771939374652819101093837... 7 is called a partial repeating decimal.  

Back in 2000 I noticed a pattern in the decimal expansions of sin(10^-n) for growing n. Here is table of some integer n:

n                sin(10^-n)

1 9.98334166*10^-2

Back in 2000 I published A034948, A036663, and A036664 in Sloane's Integer Sequences, now OEIS.

But today I decided to find the exact values of some such quotients.

1/9801=0.repeating(000

100010203040506070809101112131415161718192021222324252627282930313233

Let d be the Feigenbaum delta constant 4.66920160910299067185320382046620161..., a be the Feigenbaum alpha constant 2.50290787509589282228390287321821578... and m be the MRB constant 0.18785964246206712024851793405427323....

d*m - 2 (600 a - 2537)/(5 a - 2373) = 9.232940534412995...*10^-19

or you could write

1/10*(d*m + 564446/(2373 - 5 a)) - 24 = 9.232940534412995...*10^-20