# Could I do this to an infinite series?

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?

• 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. – cemulate Feb 1 '16 at 3:15
• @cemulate - Ahh so for this to be true, both sums have to be convergent? – Max Echendu Feb 1 '16 at 3:19
• @MaxEchendu If they are not, then the quantity $\sum_{k=1}^\infty a_k$ is not defined -- manipulating it does not really make sense. – Clement C. Feb 1 '16 at 3:36
• @ClementC. - Ahh alright, yeah my lecturer always does tell me "STOP TRYING TO DEFINE THE UNDEFINED" haha. Thanks for the reply :) – Max Echendu Feb 1 '16 at 3:39
• @cemulate you should consider posting this as an answer – Stella Biderman Feb 1 '16 at 5:08

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