I have a recurrence relation which is like the following:
$$ T(n) = 2T(n/2) + \log_2 n. $$
I am using recursion tree method to solve this. And at the end, i came up with the following equation:
$$ T(n)=(2\log_2 n)(n-1)-(1*2 + 2*2^2 + \ldots + k*2^k) $$ where $k=\log_2 n$.
I am trying to find a theta notation for this equation. But i cannot find a closed formula for the sum $$ 1*2 + 2*2^2 + \ldots + k*2^k.$$
How can i find a big theta notation for $T(n)$?