# question regarding proper mathematical form

Hi there say I have the following two sums:

$$A=\sum_{k=0}^{n}k$$

$$B=\sum_{k=0}^{n}2 \cdot k$$

where $n \rightarrow \infty$, should I write that as

$$A=\sum_{k=0}^{n} k, \quad n \rightarrow \infty$$

$$B=\sum_{k=0}^{n} 2 \cdot k, \quad n \rightarrow \infty$$

or is there a better way of writing it?

• Something called the limit. Can we evaluate this at $\infty$? Also just put it at top of sum. – marshal craft Apr 27 '17 at 16:31
• I know this happens at limit, but how do I write it properly? – no nein Apr 27 '17 at 16:32
• Aside from the comments below ($\infty$ on the top) another way of writing this would be: $$\sum_{k}k$$ This simply means "Sum over all $k$" and if no limit is defined, is a valid way of writing. – Mitchell Faas Apr 27 '17 at 16:36

$A=\sum^{\infty}_{k=0} k$ and similarly for $B$.

• I just thought in general it was bad form to write $\infty$ in a sum, is the way I wrote it wrong? – no nein Apr 27 '17 at 16:32
• Nope, not wrong, just different. – Mitchell Faas Apr 27 '17 at 16:34
• @nonein I remember it being emphasized in one of my early advanced math classes that it was actually a limit, but in the literature it's 99% going to just use $\sum^\infty$ – awright96 Apr 27 '17 at 16:35
• Yep in the early math classes we were not allowed to write it down as I did either in order to understand that $\infty$ is not a limit. But as time passed noone really cared anymore.. :D – Tesla Apr 27 '17 at 16:56

You might write

$$A_n=\sum_{k=0}^{n}k$$

and then $$A = \lim_{n \to \infty} A_n=\sum_{k=0}^{\infty}k .$$

But knowing how to write it doesn't make it right. This limit does not exist. You can't add up all the integers starting at $0$.

You might say "$A = \infty$" but that's just shorthand for the fact that the sum grows without bound. "Infinity" is not a number.

• Oh yeah I know that, it was purely for illustrative purposes, I have just seen some people write it similar to how I wrote it in articles, is that an incorrect way of doing it? – no nein Apr 27 '17 at 16:34

The full version is:

$$\lim_{n \to \infty} \left(\sum_{k=0}^n 2k\right)$$

However, we write this as $$\sum_{k=0}^{\infty}2k$$ for short.

In words, you might like to say "The limit of $$\sum_{k=0}^n 2k$$ as $n$ tends to infinity."

Something called the limit. Can we evaluate this at $\infty$? Also just put it at top of sum. $$A=\sum_{k=0}^{\infty}k=\lim_{n\to \infty} \sum_{k=0}^{n}k$$

If $\displaystyle A =\sum_{k=1}^n k$ then you could write \begin{align} & \lim_{n\to\infty} A = \text{something} \\[10pt] \text{or } & A \to \text{something as } n\to\infty \\[10pt] \text{or } & \sum_{k=1}^n k \to \text{something as } n \to\infty \\[10pt] \text{or } & \lim_{n\to\infty} \sum_{k=1}^n k = \text{something} \\[10pt] \text{or (since this is a sum) } & \sum_{k=1}^\infty k = \text{something}. \end{align}