What does this sigma convention mean? I saw this convention used on a website when referring to Backpropagation. I'm struggling to even understand what it means, can someone please help me?
$\frac{1}{2}\sum_{(k\in K)}$
I was under the impression that Sigma should always be like this:
$S = \displaystyle\sum_{k=1}^{m} 5^{k}$
Clearly my first equation is missing the top and bottom elements on the Sigma, also the convention on the website has the middle portion (in front of the sigma: $(k \in K)$) lower down on the front of the sigma as opposed to in the middle like in the second Sigma equation ($5^{k}$)
Could someone help me to understand what it means. Also what does $(k\in K)$ mean with the sigma.
Just wondering if anyone has any thoughts on this?
Thanks
 A: The usual notation for summations you see in high school are something like 
$$ \sum_{k=1}^n\frac{1}{2^k}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^n}.$$
You are summing over an index set $\{1,\ldots, n\}$. Technically, this assumes that there is an ordering on your elements, in the sense that you start with $k=1$ and terminate with $k=n$. However, we can define summation over more general sets. Suppose we wanted to get the total mass of some distinct bunches of bananas, $b_1,\ldots, b_n$. Let $M(b_k)$ be the mass (in kg) of the $k^{th}$ bunch of bananas. We might find the total mass of all the bananas by summing over the bunches in the order we gave them, $b_1,\ldots, b_n$ as 
$$ M_{total}=\sum_{k=1}^nM(b_k).$$
On the other hand, there's no need to prescribe an order to these bunches of bananas, as they have no intrinsic ordering. So, we could just express 
$$ M_{total}=\sum_{b\in B} M(b)$$
where $B$ is the set of bunches of bananas, and $M(b)$ denotes the mass as before. The point of this silly example is to show that for more general sets we can define sums of values attributed to each element of the set. More generally, given a (finite) set $X$ and a function $f:X\to \mathbf{R}$, it makes perfect sense to take 
$$ \sum_{x\in X}f(x).$$
This returns a real number. As for the notation $\sum_{x\in X}$, this is just a different formatting of the same expression.
