Why do we need three indices for Christoffel Symbols I read the following results on covariant differentiation (summation convention applied): If $X=X^i e_i$ and $Y=Y^je_j$, then
$$\nabla_XY = X^i\nabla_{e_i}(Y^je_j) = X^ie_i(Y^j)e_j + X^iY^j\Gamma_{ij}^ke_k = [X(Y^k)+X^iY^j\Gamma_{ij}^k]e_k.$$
I do not understand the usage of the third index $k$ here. Where does the third index come from, i.e. the $k$ in $\Gamma_{ij}^k$, please? Thank you! Consider the case where $n=2$. Could anyone explain how to fill the question marks, please? I think my question is where the third index come from in the definition of Christoffel symbols, i.e. $\nabla_{e_j}e_k =\Gamma_{jk}^i e_i$. Why do we need a third index $k$ for it?
$$\begin{align}\nabla_XY &= \nabla_{X^1e_1+X^2e_2}(Y^1e_1+Y^2e_2) \\ &= X^1\nabla_{e_1}(Y^1e_1) + X^1\nabla_{e_1}(Y^2e_2) + X^2\nabla_{e_2}(Y^1e_1) + X^2\nabla_{e_2}(Y^2e_2) \\ & =  X^1e_1(Y^1)e_1 + X^1e_1(Y^2)e_2 + X^2e_2(Y^1)e_1 + X^2e_2(Y^2)e_2 \\ &+ X^1Y^1\nabla_{e_1}e_1 + X^1Y^2\nabla_{e_1}e_2 + X^2Y^1\nabla_{e_2}e_1 + X^2Y^2\nabla_{e_2}e_2 \\ &= \sum_{i=1}^2 X(Y^i)e_i + \sum_{i,j=1}^2 X^iY^j\nabla_{e_i}e_j\\&= \cdots??? \\ &= [X(Y^k)+X^iY^j\Gamma_{ij}^k]e_k\end{align}$$
 A: By definition, $\nabla_{e_i}e_j = \Gamma_{ij}^k e_k $, again using the summation convention. The left hand side is a vector and you're just expanding it in the basis vectors, with the Christoffel symbols being the coefficients.
Maybe it would help you to not omit the summation: $\nabla_{e_i} e_j = \sum_k \Gamma_{ij}^k e_k$. Let's say instead of having some compound expression on the left side, you had some vector $v$. For simplicity, let's take the dimension to be $2$, as in your question. You want to  write $v$ in terms of the basis vectors, $e_1,e_2$. Then $v = v_1e_1+v_2e_2$. In general, you write $v = v_1e_1+v_2e_2 + \dots v_ne_n$, or in short, $v = \sum_{k=1}^n v_ke_k$.
To be even briefer, one omits the summation and just says $v = v_ke_k$. Do you understand why we needed an extra index $k$ on the right hand side here, even though there weren't any on the left? Now let's go back to the original expression. Instead of having a fixed vector $v$, you have a bunch of vectors $v_{ij}$, given by $\nabla_{e_i}e_j$. To each of them you  can apply the same procedure and expand in terms of $e_k$. Except now the coefficients will depend not only on $k$ but also $i$ and $j$. So this number which depends on these three indices is what you call $\Gamma_{ij}^k$.
A: The Christoffel symbols are defined so that for all $ i $ and $ j $
$$
\nabla_{e_i} e_j = \Gamma_{ij}^{k} e_k 
$$
This term comes up because of the product rule:
$$
\nabla_{e_i} (Y^j e_j) = (\nabla_{e_i} Y^j)e_j + Y^j (\nabla_{e_i} e_j )
$$
