How can the number e be irrational (follows no pattern) if its Taylor Series representation is constructed using a beautiful pattern?

Somewhere in your math education, someone gave you the idea that "irrational" means "follows no pattern". This is false. Unfortunately it is a very common misconception, perpetuated by teachers who are either confused or choose to make vague statements in an attempt to make things informal or conversational.

Answer by Alon Amit:

Somewhere in your math education, someone gave you the idea that "irrational" means "follows no pattern". This is false. Unfortunately it is a very common misconception, perpetuated by teachers who are either confused or choose to make vague statements in an attempt to make things informal or conversational.
A number is rational if it is the ratio of integers. [math]\frac{23}{17}[/math] is rational. So is [math]2016[/math].
A number is irrational if it's not rational. That's all there is to it.
When you consider the decimal expansion of real numbers, you observe that rational numbers are distinguished by having a decimal expansion that is eventually repeating. Note the term "repeating". It means the expansion goes something like
[math]0.\mbox{whatever}\mbox{repeat}\mbox{repeat}\mbox{repeat}\ldots[/math]
The "eventually" part says that there may be some initial digits that aren't part of the repeating cycle, and then you have a repeating cycle.
[math]\frac{1}{3}=0.333333333\ldots[/math]
[math]\frac{1}{7}=0.142857142857142857\ldots[/math]
[math]\frac{511}{49950}=0.010230230230230\ldots[/math]
So again: rational means that the decimal expansion is eventually repeating.  It doesn't mean that the number itself "follows a pattern". Numbers don't have or lack patterns. Numbers can be represented in many ways, some simple, some complex, some repeating, some non-repeating, some patterned, some not. Being irrational is one specific, precise property of numbers, and it doesn't mean "follows no pattern". And decimal expansions can have all kinds of patterns which aren't simply "repeating".
Look at the number
[math]0.19119111911119111119\ldots[/math]
The decimal expansion has one 1, then 9, then two 1's, then 9, then 3 1's, then 9, then 4 1's, then 9, and so on. Notice the pattern? It's an easy pattern. You can figure out how this expansion continues forever. But the number is irrational, because the pattern isn't a simple repeating cycle.
Many numbers are defined by infinite series:
[math]\displaystyle x = \sum_{n=1}^\infty \frac{1}{n^2}[/math]
[math]\displaystyle y = \sum_{n=1}^\infty \frac{1}{2^n}[/math]
[math]\displaystyle z = \sum_{n=1}^\infty \frac{n}{2^n}[/math]
[math]\displaystyle w = \sum_{n=1}^\infty \frac{1}{n 2^n}[/math]
[math]\displaystyle e = \sum_{n=0}^\infty \frac{1}{n!}[/math]
All of those series are highly patterned: they use a simple expression that is added to itself over and over again with different values of the variable [math]n[/math]. But this is not a decimal expansion, and this is not a simple cycle which repeats. Therefore, the mere existence of such a "patterned series" has nothing to do with whether the infinite sum is rational or irrational
In our examples, the number [math]x=\frac{\pi^2}{6}[/math] is irrational. The number [math]y[/math],  whose series looks almost exactly the same, is rational: in fact, it's just [math]y=1[/math]. The number [math]z[/math] is just 2, also rational. The number [math]w[/math], which has an almost identical series to [math]z[/math] (we just moved the [math]n[/math] from the numerator to the denominator), is irrational. It is [math]\ln(2)[/math].
(By the way, none of these is a "Taylor expansion". This is the wrong term. Taylor series apply to functions, not individual numbers.)
The decimal expansion of [math]e[/math] exhibits no obvious pattern:
[math]e = 2.71828182845904523536\ldots[/math]
And indeed, as a sequence of numbers between 0 and 9, it is a pretty random-looking sequence. However, it is a sequence that is defined by a fairly simple rule: you can write a very short computer program that spits out the digits of [math]e[/math] until it hits the physical limits of memory and capacity of your computer. One of the most reasonable ways to define "pattern" is "something that is the output of a short computer program".
Other ways of representing [math]e[/math] exhibit more obvious patterns. For example, the continued fraction representation of [math]e[/math] is
[math]e = [2; 1,2,1,1,4,1,1,6,1,1,8,\ldots][/math].
So in summary, [math]e[/math] has a simple-looking continued fraction, a simple-looking series representation, and a non-simple-looking decimal expansion. The inherent information content of all of those representations is the same, so they are all just as "patterned", though some of those patterns are easier to comprehend. None of those patterns has any direct relationship with being rational or irrational.

How can the number e be irrational (follows no pattern) if its Taylor Series representation is constructed using a beautiful pattern?

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