Given $S_n=a_1+a_2+a_3+\cdots+a_n$, both $\lim\limits_{n\to\infty}{a_n}$ and $\lim\limits_{n\to\infty}{S_n}$ exists.
Is the following equation correct? If not, give a counter example please.
$\lim\limits_{n\to\infty}S_n=\lim\limits_{n\to\infty}{a_1}+\lim\limits_{n\to\infty}{a_2}+\lim\limits_{n\to\infty}{a_3}+\cdots+\lim\limits_{n\to\infty}{a_n}$
What you've written doesn't make sense: $$\lim_{n\to\infty}\sum_{i=1}^n a_i=\sum_{i=1}^n\lim_{n\to\infty} a_i.$$ Now you have an instance of $n$ occurring outside the limits (namely, the upper bound of the sum), which is an illegal move as far as formal manipulation is concerned. You may as well use it on other limits like $\lim_{n\to\infty}n=n\lim_{n\to\infty}1=n$ and get nonsensical answers. I believe we say that $n$ is a "dummy variable," because it is a variable that gets momentarily introduced and then eradicated afterwards when making an evaluation, much like defining a local variable in a programming subroutine.
What do you mean by $\lim\limits_{n\to\infty} a_1$, $\lim\limits_{n\to\infty} a_2$, etc.? The usual meaning would be that you are considering constant sequences, in which case you would simply have $\lim\limits_{n\to\infty} a_1=a_1$, $\lim\limits_{n\to\infty} a_2=a_2$, etc. However, even then, the right-hand side of your last equation does not have clear meaning. $n$ is being overloaded.
– Jonas Meyer Sep 15 '11 at 04:13