So if an infinite series converges to any real than the limit of the partial sums sequence is equal to $0$, right?
Does that mean that the limit of the series is $0$ too?
What confuses me is "is the limit of an infinite series equal to it's sum?"
so if the limit of the partial sums sequence is $0$, then the sum of the sequence is $0$, which allows me to say that the limit of the series is $0$, so the series converges to $0$?
Best Answer
By definition, the sum of an infinite series is the limit of its partial sums:
$$\sum_{i = 0}^{ \infty}a_i := \lim_{N \to \infty}\sum_{i = 0}^Na_i$$
Now, in order for this limit to converge to a real number $L$, it would certainly have to be that the distance between each partial sum and the next gets arbitrarily small - after all, that's a necessary condition for any limit to converge.
$$\sum_{i = 0}^{N + 1}a_i - \sum_{i = 0}^Na_i = a_{N + 1}$$
So the values of $a_N$ must go to zero. That means that it is not correct to say that the limit of the partial sums must be zero, but it is correct to say that the limit of the terms of the series must be zero.
There is a point of confusion here, though. You keep referring to the limit of a series. That's not typical terminology, because it isn't clear whether you mean the limit of the terms (because you're not mentioning adding things up) or the limit of the partial sums (because you're calling it a "series" and you usually sum series). Generally, we say limit of a sequence and sum of a series. So, to put things as precisely as I can:
Let $a_0, a_1, \ldots, a_i, \ldots$ be an infinite sequence. The limit of the sequence is $\lim_{i \to \infty}a_i$. The sum of the series is $\sum_{i = 0}^{\infty}a_i = \lim_{N \to \infty}\sum_{i = 0}^Na_i$. If the sum of the series converges to a real number, then the limit of the sequence must be zero. Because the sum of the series may be nonzero, the limit of the sequence is not necessarily the same as the sum of the series.