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.

{% img center http://farm6.staticflickr.com/5504/12227941034_4ce9d69db4_z_d.jpg %}

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

it.

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

{% img center http://farm6.staticflickr.com/5535/12227532975_ee70979b52_z_d.jpg %}

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

!!