Can a transcendental number like $e$ or $\pi$ be compressed as not algorithmically random?

Question

The related and interesting fields of Information Theory, Turing Computability, Kolmogorov Complexity and Algorithmic Information Theory, give definitions of algorithmically random numbers.

An algorithmically random number is a number (in some encoding, usually binary) for which the shortest program (e.g using a Turing Machine) to generate the number, has the same length (number of bits) as the number itself.

In this sense numbers like $\sqrt{e}$ or $\pi$ are not random since well known (mathematical) relations exist which in effect function as algorithms for these numbers.

However, especially for $e$ and $\pi$ (which are transcendental numbers) it is known that they are defined by infinite power series.

For example $e = \sum_{n=0}^\infty \frac{1}{n!}$

So even though a number, which is the binary representation of $\sqrt{e}$, is not alg. random, a program would (still?) need the description of the (infinite) bits of the (transcendental) number $e$ itself.

Can transcendental numbers (really) be compressed?

Where is this argument wrong?

UPDATE:

Also note the fact that for almost all transcendental numbers, and irrational numbers in general, the frequency of digits is uniform (much like a random sequence). So its Shannon entropy should be equal to a random string, however the Kolmogorov Complexity, which is related to Shannon Entropy, would be different (as not alg. random)

Thank you