Proof for the product of an infinite series

Question

Taking for example 2 x 2 x 2 ..., I know that we can represent this as r = 2r, which should result to be r = 0.

Now my question arises when attempting to prove whether the (k+1)^th element is greater than the k^th element, i.e.

2^k+1 = 2^k x 2, which should suggest 2^k+1 > 2^k, thus by induction the product should continuously increase by a product of 2, i.e. r > 2, but how is the result r = 0 justified?

Answer 1

Taking for example 2 x 2 x 2 ..., I know that we can represent this as r = 2r, which should result to be r = 0.

Only if the product actually converges. For a sequence of numbers $a_n$, we say

$$\prod_{i=1}^\infty a_n \text{ converges if and only if } \lim_{n \to \infty} \prod_{i=1}^n a_n \text{ exists}$$

Moreover, the value attributed to the infinite product is given by that limit if it exists.

Let $a_n = 2$ for all positive integers $n$. Then $\prod_{i=1}^n a_n = 2^n$. Obviously, this diverges as $n \to \infty$, and thus so does the infinite product. However, if $a_n = 1/2$ for example, then you can show the infinite product is zero.

Clearly, the claim $r=0$ is totally unjustified in this light. Such operations are not always justified.