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) L₀ . You will find that f(x) quickly approaches 0. This obviously doesn't give us e, the actual limit of that function.

© Scott Sherman 1999