Is it appropriate to use $\lim$ for a finite sum?

Question

Is it appropriate to use $\lim$ for a finite sum?

I have a sequence; say the partial sum to the $n^{th}$ term is given by $f^n(x)$

Suppose it converges to some value $y$:

$\lim_{n\to\infty}f^n(x)=y$

Now suppose I know that it converges to that value in a finite number of steps, i.e. there exists some $m\in\Bbb N$ such that $f^m(x)=y$

Is it still acceptable to write $\lim_{n\to\infty}f^n(x)=y$ or is it necessary to state something like $\exists m\in\Bbb N\mid f^n(x)=y\forall n\geq m$?

Answer 1

You can still say the limit is $y$ as long as the tail of the sequence (from $m$ on) has limit $y$. That will be the case if the tail is constantly $y$.

Answer 2

Yes. For example, the sequence (5,4,3,2,2,2,2,2,…), which is eventually constant, has limit 2. Thus, if we define $a_n$ to be the $n$th element of that sequence, then we have $\lim_{n\to\infty}a_n=0$.

Maybe your intuition was that a limit describes something you "get closer and closer to but never reach." But that's false; you're absolutely allowed to reach it. Some sequences do, some sequences don't.

Answer 3

Let us see in general the definition of a limit of a sequence (a finite/infinite sum can surely be seen as a sequence, so we will stick to this):

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers and let $a\in\mathbb{R}$. We will say that $a_n$ converges to $a$ and write: $$\lim_{n\to\infty}a_n=a_n$$ if and only if $$\forall\ \epsilon>0\ \exists\ n_0=n_0(\epsilon)\in\mathbb{N}:|a_n-a|<\epsilon\ \forall\ n\geq n_0$$

As you see, there is no need to consider whether $a_n$ is finally constant or not, since, what is requested, by definition, is that, finally, $a_n$ gets as close as possible (for every $\epsilon>0$) to $a$, regardless whether this might be trivial or not.

Hope this helped! :)