question regarding proper mathematical form

Question

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?

Answer 1

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

Answer 2

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.

Answer 3

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."

Answer 4

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

Answer 5

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}