### Infinite Series and Sequences

One caution to keep in mind when using leftins is that infinite series and
sequences are not well behaved. The most obvious example is to take a leftin which
is equivalent to a real number (other than a positive integer) and sum up its digits
as you would for the decimal representation of a number. Clearly 9 + 9*10 + 9*100 +
9*1000 + ^{...} does not equal -1.

One way to take this is to pretend like we're a quantum physicist: if one
technique generates infinity, rephrase it until the infinities go away.
Unfortunately, this doesn't give us consistent results.

For example, one method of turning the digits of a leftin into an infinite sum
that converges is to use the following series:

...(1) -1
...(010) -10/999
...(00100) -100/99999
...

It's obvious that you can turn the digits of a leftin into a converging infinite
series using that series. It's also obvious that this will only generate
numbers between -1 and -1.1.

Limits are also out of bounds. For example, you may try to find *e* by
finding the limit as x approaches infinity of f(x) = (1 + 1/x)^{x}.
If you try 9, 99, 999, etc., you can see that f(x)
*L*_{0} . You will find that f(x) quickly approaches 0.
This obviously doesn't give us *e*, the actual limit of that function.

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© Scott Sherman 1999