I have a simple idea that I would like to make into a formula

Question

If I wanted to write one plus one, recurring, equals infinity....

Does this make sense?

If so, how would it be written as a formula?

Answer 1

This depends on how strictly rigorous you want to be. You will, fairly often, see statements such as $$\sum_{i=0}^\infty 1=\infty$$

When this is written the correct interpretation should not really be that the sum of infinitely many $1$'s is infinity. Rather it should be that the limit of partial sums diverges to positive infinity, or written mathematically $$\lim_{n\to\infty}\sum_{i=0}^n1=+\infty$$

There is a very good reason to try to always stay very rigorous when dealing with potential infinites (and even with actual infinities but surprisingly they seem to be less of an issue). That reason is that unless you have a lot of experience and training in mathematics chances are, roughly one out of two statements or ideas you will have about potential infinities, will not only be wrong, but usually won't even make good sense.

Answer 2

An alternative way to look at it might be the limit of a recursive function. $$ f(0) = 1 $$ $$ f(n) = f(n-1) + 1 $$ $$ \lim\limits_{n \to \infty} f(n) = \infty $$

Answer 3

You can say like...

$1 + 1 + 1 ... = \sum\limits_{i=1}^{\infty} 1 = +\infty$

Though this is simply the notation.