1
$\begingroup$

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.

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

  • 3
    What do you mean by $\lim_{n\to\infty}a_1$? $a_1$ does not depend on $n$.2011-09-15
  • 3
    Correct/Incorrect can only be said about meaningful statements. Your statement makes no sense.2011-09-15
  • 0
    If $\lim\limits_{n\to\infty} S_n$ exists, then $\lim\limits_{n\to\infty} a_n =0$. 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.2011-09-15
  • 1
    Somewhat related: http://math.stackexchange.com/questions/59795/proof-of-1-0-by-mathematical-induction2011-09-15
  • 1
    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

1 Answers 1