Proof of the rationality of $e + \pi$ via their series representation?

Question

I found just one question similar to this, but it had been edited, so hopefully this isn't asked too often.

Given the formulas via infinite sums for expressing $e$ and $\pi$... $$ e = \sum_{n=0}^\infty \frac{1}{n!} = 1+ 1 + \frac{1}{2} + \frac{1}{6} + ...$$ $$ \pi = 4\sum_{n=0}^\infty \frac{(-1)^{n}}{2n+1} = 4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7} + ... $$

Since every number in each series is rational, and the sum of two rational numbers is also rational, doesn't that imply that $e$ + $\pi$ is rational?

I realize this may seem like a simple argument, but the only way I can imagine it being wrong is if there is a principle I am missing regarding adding two infinite sums.

Answer 1

An infinite sum of rationals may be either rational or irrational. In fact, any real number, whether rational or not, is an infinite sum of rationals $$n+\frac{d_1}{10}+\frac{d_2}{100}+\frac{d_3}{1000}+\cdots\ :$$ this is exactly what its decimal representation means.

The problem is that an "infinite sum" is not, strictly speaking, a sum. It is a sum together with a limit: by definition, $$\sum_{n=1}^\infty a_n$$ means $$\lim_{N\to\infty}\sum_{n=1}^N a_n\ .$$ The fact that a limit is involved changes a number of the usual rules regarding addition.

Answer 2

Using your arguments, $e$ and $\pi$ are rational. Are they?

Answer 3

No, that argument doesn't work. What it does establish is that $e+\pi$ can be written as an infinite sum of rational numbers - however, this is true of every number (since, for instance, $\pi=3+.1+.04+.001+\ldots$). However, this does not mean that $e+\pi$ is rational - in fact, your argument would necessarily also prove that $e$ and $\pi$ are also both rational, especially if you consider that such a series exists for $\frac{\pi}2$ and $\frac{\pi}2+\frac{\pi}2=\pi$.

Answer 4

By the same reasoning, you could conclude that each of these series individually converges to a rational, which is false.

It is indeed correct that a sum of rationals is always rational. The problem is that an infinite series is not a sum. This is a trap that many fall into. Even experienced mathematicians come to think of infinite series as a sums, but they simply are not. They may often be treated that way, but, in fact they are limits, not sums.

The symbol $$S=\sum_{n=1}^{\infty} a_n$$ is shorthand notation for $$S=\lim_{k\to\infty}s_k$$ where $$s_k=\sum_{n=1}^k a_n$$

Each $s_k$ is a genuine sum (called the $k$th partial sum), but $S$ is not a sum. $S$ is a limit -- a limit of partial sums.

Example: In fact, you can't just write down an irrational number using plain decimal notation. Suppose you want to write down pi. You could write "3.141", or "3.14159", or "3.14159265", but these are all just rational numbers that are getting closer and closer to pi. None of them is pi. You typically indicate the limit pi by including an ellipsis, as "3.14159265...".