Bound/free variables in $\sum_{k=1}^{10}f(k,n)$ I'm looking at this treatment of bound and free variables. Down a bit is this
$$\sum_{k=1}^{10}f(k,n) $$
but then the cryptic explanation

$n$ is a free variable and $k$ is a bound variable; consequently the
value of this expression depends on the value of $n$, but there is
nothing called $k$ on which it could depend.

Can anyone tell me what they're saying here? If $f(k,n) = (k + n)$, i.e.,
\begin{align}
\sum_{k=1}^{3}(k+n) &= (1+n) + (2+n) + (3+n) \\
&= (1+2+3) + (n+n+n) \\
&= 6 + 3n
\end{align}
It's obvious that $n$ is an unknown variable from elsewhere throughout summing over $k$, but I don't get the wording of the above quote.
 A: 
It's obvious that $n$ is an unknown variable from elsewhere throughout summing over $k$, but I don't get the wording of the above quote.

That's it.
In the expression, $n$ is free to take on values from elsewhere.  The expression is thus dependent on what value is assigned to $n$ there.
However, $k$ is bound to take the values designated by the series.  The term following the series quantifier is to be evaluated for each value of $k$, and these values are then added together.   Further, any $k$ referrenced inside this scope is only this bound variable, and not any mention of $k$ outside the expression.
In programming terminology, $k$ would be called a local variable or loop iterator, if that helps, while $n$ would be a global variable.
A: It seems quite confusing in this context because, as you've identified, $f(k,n)$ does depend on the first variable, which we seem to have represented by $k$. The distinction is, intuitively, that although the input associated to $k$ "matters", $k$ just encodes the directions "if we choose some $n$, then add $f(1,n)$, $f(2,n)$, and $f(3,n)$", that we represent by a sum that happens to use the variable $k$ for indexing. More precisely, it is being "quantified over", such that $\sum^3_{k=1}f(k,n)$ says "add $f(k,n)$, for all $1\leq k \leq 3$". We are not free to "choose a specific value for $k$", as $k$ is already bound, rather "assigned", to vary over the values $k=1$, $k=2$, and $k=3$.
Instead, imagine that we have another function, $g:\mathbb{N}\rightarrow \mathbb{R}$, defined such that $g(n)=\sum^3_{k=1}f(k,n)$. This gives our original operation but only depends on $n$. Note, if we chose instead $g(n)=\sum^3_{m=1}f(m,n)$, it would be the exact same operation, as the variable $m$ or $k$ is not important, as its directions are the same either way, "sum over all values $m/k=1$, $m/k=2$, and $m/k=3$".
A: Perhaps the distinction is easier to understand in a computer programming context.
def f(x, y):
    return x + y

def sum_f(n):
    return sum(f(k, n) for k in range(1, 11))

Within sum_f, n is a parameter passed to the function, but $k$ is just a local variable that has meaning only within the sequence being summed.  Unless you're looking at the source code for sum_f, you'd have no idea what k is.  Its name is just an implementation detail.  You can say that it's bound to that one expression, whereas the caller is free to specify a value for n.
A: I fully understand (mainly from programming) what is meant by a bound and free variable -- which is why this was so maddening. I think the it "...on which it could depend" simply means the $k$ is not part of $n$'s definition, has no connection to $n$'s definition term, wherever it may be defined extra-summation. Hence, I'm overreacting. Alas, still working through math panic issues....
