I'm work through some questions relating to connection coefficents. My question is more about the summation notation being used. Why is there no starting point (or end point) for the summation here? For the i, j, k, I take these to range from 1 to 2 for a 2 dimensional surface. Does this just imply that l does likewise?
$$ \Gamma^i_{jk} = \sum_{l} g^{il} \left( \dfrac{\partial g_{lk}}{\partial x^j} + \dfrac{\partial g_{jl}}{\partial x^k} - \dfrac{\partial g_{jk}}{\partial x^l} \right) $$
On
Another example of use of a similar notation which might make more sense to you is
$$\sum_{n\in \mathbb{N}}n=1+2+3+\dots$$
If the set is not specified directly under your sum, then $l$ probably runs over some set of values which make sense, maybe try reading back, that set was probably stated either explicitly or implicitly
This formula comes from differential geometry, it takes place in the Cartesian coordinate space $\mathbb{R}^n$ for some value of the dimension $n$. Once $n$ has been specified, the indices $i,j,k,l$, by convention, take values in the index set $\{1,...,n\}$.
In this formula, the values $i,j,k$ are fixed and $l$ runs freely over the whole index set. So the summation symbol $\displaystyle\sum_l$, by convention, is an abbreviation for $$\sum_{l=1}^n $$ Whatever differential geometry text you read, there are likely to be hundreds of such summations, and there will be conventions for writing them in abbreviated forms.