The usual convention is that
$$ \sum_{n=1}^{N} a_n $$
is the sum of all $a_n$ with indices between the lower and upper limits of summation. Hence
$$ \sum_{n=1}^{N} a_n = \sum_{1\le n \le N} a_n = \sum_{n=1}^{\lfloor N \rfloor} a_n = a_1 + a_2 + \dotsb + a_{\lfloor N \rfloor}. $$
That being said, if you are interested in what happens as $N$ goes to infinity (as is often the case), it doesn't really matter which convention you choose.
The following is probably overkill, but the above can actually be justified in a meaningful way via a branch of mathematics called measure theory: a series is a special kind of integral. Indeed, if you know any measure theory, a series is an integral of the form
$$ \sum_{n=1}^{N} a_n = \int_{[0,N]} a(x) \,\mathrm{d}\mu(x), $$
where $a : \mathbb{R} \to \mathbb{R}$ (or some other codomain) such that $$ a(x) = \begin{cases} a_n & \text{if $x = n \in \mathbb{N}$, and} \\ 0 & \text{otherwise}, \end{cases} $$
and $\mu$ is counting measure on $\mathbb{N}$. If $N$ is not an integer, then
\begin{align}
\sum_{n=1}^{N} a_n &= \int_{[0,N]} a(x)\,\mathrm{d}\mu(x) && \text{(by definition)} \\
&= \int_{[0,\lfloor N \rfloor]} a(x) \, \mathrm{d}\mu(x) + \int_{(\lfloor N\rfloor, N]} a(x) \,\mathrm{d}\mu(x) && \text{(additivity of integrals)} \\
&= \int_{[0,\lfloor N \rfloor]} a(x) \, \mathrm{d}\mu(x) \tag{1}\\
&= \sum_{n=1}^{\lfloor N \rfloor} a_n. && \text{(definition)}
\end{align}
The only mystery here is at (1). Here, we are using the fact that the measure of an interval which contains no integers must be zero. That is,
$$ \mu((\lfloor N \rfloor, N]) = 0. $$
In a measure space, integration over a set of measure zero is always zero. Hence
$$ \int_{(\lfloor N\rfloor, N]} a(x) \,\mathrm{d}\mu(x) = 0. $$
In short, the identity above is justified, that is
$$ \sum_{n=1}^{N} a_n = \sum_{n=1}^{\lfloor N \rfloor} a_n. $$
