# mathematica 1 slash 9998

Jan 30, 2014

I saw this piece about how `1/1998` was approximated by series of powers of `2`, which raised the question, “Can Mathematica help confirm this?” (of course it can!)

First, a gut check to see the decimal expansion

``````N[1/9998, 100]
``````

which shows

``````0.00010002000400080016003200640128025605121024204840968193638727745549 \
10982196439287857571514302860572114
``````

(if you don’t specify the `100` in the argument to `N[]`, it will show you just `0.00010002` by default)

This can be visualized as

``````0.0001 +
0.00000002 +
0.000000000004 +
0.0000000000000008 +
0.00000000000000000016 +
...
0.000000000000000000000000000000000256 +
...
``````

All right, so we do see these powers of 2! They seem to be in “groups” – `0004`, `0016`, `1024` and so on.

So we want to check if something like `2^i/10000^i` can add up to this number.

``````Parallelize[
Table[1/9998 - Total[Table[2^i/10000^i, {i, 1, n}]], {n, 1, 10000}] ]
``````

I initially tried this without `Parallelize[]`, but that kept going for a while, so I had to abort it (`Evaluation`->`Abort Evaluation`) and try this version instead.

Anyway, so this whirs for a while. As this `htop` output shows, it really does use as many cores as possible.

Eventually when it’s done, we can `ListPlot` it.

``````ListPlot[%118, Frame -> True, FrameStyle -> Black,
PlotRange -> { {1, 10000}, {-0.000105, -0.000095} }]
``````

As this graph shows, the result is remarkably close to a constant value of `-0.0001` !!