Summation of a term to infinity

It's not only valid, it's how it's defined.

---

Note that the operation "addition" is defined only if we apply it a finite amount of times. Thus, adding an infinite amount of terms doesn't make sense. We'll have to define it as a limit, as that only includes nice, finite sums.

This is the definition of infinite series. It is the limit of the partial sums $S_n$:

$$S_n = \sum_{k = 0}^n a_k$$

$$\sum_{k = 0}^{\infty} a_k := \lim_{n \to \infty} S_n = \lim_{n \to \infty} \sum_{k = 0}^n a_k$$