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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.

$\lim\limits_{n\to\infty}S_n=\lim\limits_{n\to\infty}{a_1}+\lim\limits_{n\to\infty}{a_2}+\lim\limits_{n\to\infty}{a_3}+\cdots+\lim\limits_{n\to\infty}{a_n}$

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    I wonder if the question isn't supposed to be about doubly infinite sequences, $S_n=a_{1,n}+\cdots+a_{n,n}$, $\lim_{n\to\infty}a_{i,n}$ existing for all $i$, etc.2011-09-15

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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.