For $ \overrightarrow{V} = V^i \overrightarrow{e_i} = V_i \overrightarrow{e^i} $,
$ \frac{\partial \overrightarrow{V}}{\partial x^j} = \frac{\partial V^i}{\partial x^j} \overrightarrow{e_i} + V^i \frac{\partial \overrightarrow{e_i}}{\partial x^j} = \frac{\partial V^i}{\partial x^j} \overrightarrow{e_i} + V^i (\Gamma^k_{ij} \overrightarrow{e_k}) $
Yes, I get it until here. But I don't understand how are we able to switch dummy indices i and k only for the second term of the last side so it becomes $ \frac{\partial V^i}{\partial x^j} \overrightarrow{e_i} + V^k (\Gamma^i_{ki} \overrightarrow{e_i}) = (\frac{\partial V^i}{\partial x^j} + V^k \Gamma^i_{kj}) \overrightarrow{e_i} $. Shouldn't i in the first term also be changed to k?
You have a double sum $$ \sum_i V^i \sum_k \Gamma^k_{ij} e_k = \sum_{i,k} V^i\Gamma^k_{ij}e_k $$ and if you relabel the dummy variables $i,k$ as, lets say, $\alpha,\beta$, you end up with $$ \sum_i V^i \sum_k \Gamma^k_{ij} e_k = \sum_{\alpha,\beta} V^{\alpha}\Gamma^{\beta}_{\alpha j}e_{\beta} $$ which now becomes, by relabelling $\alpha,\beta$ as $k,i$, $$ \sum_i V^i \sum_k \Gamma^k_{ij} e_k = \sum_{i,k} V^k\Gamma^i_{kj}e_i. $$ Hence, \begin{align} \sum_i \left( \partial_jV^i e_i + V^i\sum_k\Gamma^k_{ij}e_k \right) &= \left(\sum_i\partial_jV^ie_i\right) + \left(\sum_iV^i\sum_k \Gamma^k_{ij}e_k\right) \\ &= \left(\sum_i \partial_jV^ie_i\right) + \left(\sum_{i,k} V^i\Gamma^k_{ij}e_k\right) \\ &= \left(\sum_i \partial_jV^ie_i\right) + \left(\sum_{i,k}V^k\Gamma^i_{kj}e_i \right) \\ &= \left(\sum_i \partial_jV^ie_i\right) + \left(\sum_i\sum_kV^k\Gamma^i_{kj}e_i\right)\\ &= \sum_i\left( \partial_jV^i + \sum_{k}V^k\Gamma^i_{kj}\right)e_i. \end{align}
If you don't understand, look at an easy example with only two terms.