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If a had two series like so:

$$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} i + \sum^{\infty}_{k=1} k $$

Is it logical for me to say:

$$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} i = \sum^{\infty}_{i=1} i $$

Therefore:

$$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} i + \sum^{\infty}_{k=1} k $$

$$=$$

$$ \sum^{\infty}_{k=1} k + \sum^{\infty}_{k=1} k $$

$$=$$

$$ \sum^{\infty}_{k=1} 2k$$

Is this wrong?

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    $\begingroup$ Typically, we define $\sum_{k=1}^{\infty} x_k$ as $\lim_{n \rightarrow \infty} \sum_{k=1}^n x_k$ if and only if the limit exists, so this doesn't quite work. It would be correct to say that the partial sums are equal to the resulting expression, i.e. $\sum_{k=1}^n k + \sum_{k=1}^n k = \sum_{k=1}^n 2k$, but neither of these sums converge so we can't really talk about their limits. $\endgroup$ – cemulate Feb 1 '16 at 3:15
  • $\begingroup$ @cemulate - Ahh so for this to be true, both sums have to be convergent? $\endgroup$ – Max Echendu Feb 1 '16 at 3:19
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    $\begingroup$ @MaxEchendu If they are not, then the quantity $\sum_{k=1}^\infty a_k$ is not defined -- manipulating it does not really make sense. $\endgroup$ – Clement C. Feb 1 '16 at 3:36
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    $\begingroup$ @ClementC. - Ahh alright, yeah my lecturer always does tell me "STOP TRYING TO DEFINE THE UNDEFINED" haha. Thanks for the reply :) $\endgroup$ – Max Echendu Feb 1 '16 at 3:39
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    $\begingroup$ @cemulate you should consider posting this as an answer $\endgroup$ – Stella Biderman Feb 1 '16 at 5:08
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Typically, we define $$ \sum_{k=1}^{\infty} x_k = \lim_{n \rightarrow \infty} \sum_{k=1}^n x_k $$

if and only if the limit exists. So, if the limit does not exist (i.e., the sum is not convergent, then the symbol $$ \sum_{k=1}^{\infty} x_k $$

is not defined and manipulating it algebraically makes no sense.

So, your algebra doesn't really work, but however it would be correct to say: $$ \sum_{k=1}^n k + \sum_{k=1}^n k = \sum_{k=1}^n 2k $$

That is, the partial sums are additive, but neither series converges so we can't really talk about their limits/infinite sums.

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  • $\begingroup$ I wouldn't say "if and only if the limit exists": if it doesn't, then the sum doesn't exist either. $\endgroup$ – YoTengoUnLCD Feb 1 '16 at 5:22
  • $\begingroup$ Very well explained. Thank you :). $\endgroup$ – Max Echendu Feb 1 '16 at 9:46

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