Bishal Banjara said:
I felt a bit confusion here, regarding the expression of Christoffel's and other components.
I find it helpful to write out the sums explicitly. Thus$$\begin{eqnarray*}
0&=&\partial_aT^{ab}+\Gamma^a_{ad}T^{db}+\Gamma^b_{ad}T^{da}\\
&=&\sum_a\partial_aT^{ab}\\
&&+\sum_a\sum_d\Gamma^a_{ad}T^{db}\\
&&+\sum_a\sum_d\Gamma^b_{ad}T^{da}\end{eqnarray*}$$where there is no implied summation over repeated indices in the second line (there is in the first). There is one free index, so this us four equations, one for each ##b##. Doing the ##b=r## case, you have$$\begin{eqnarray*}
0&=&\sum_a\partial_aT^{ar}\\
&&+\sum_a\sum_d\Gamma^a_{ad}T^{dr}\\
&&+\sum_a\sum_d\Gamma^r_{ad}T^{da}\\
&=&\partial_rT^{rr}\\
&&+\sum_a\Gamma^a_{ar}T^{rr}\\
&&+\sum_a\Gamma^r_{aa}T^{aa}\end{eqnarray*}$$Again there is no implied summation. We have used the fact that ##T^{ab}=0\ \forall\ a\neq b## to drop summation terms that are obviously zero in order to go from the first line to the second.
I should note that putting the summation signs in has absolutely no effect except that I personally find it useful as a way of tracking what is being summed over and what is not.