In the 4th edition of his Data Structures textbook, Weis gives a proof of part of the Master Theorem. This proof says "Let us ... ignore the constant factor in $\theta(N^k)$ ... I don't understand why it ignoring that constant is valid. (I know that the real Master Theorem is broader than the version presented in the text. I'm just having trouble understanding the argument presented in the text.)
Specifically, the relevant part of Theorem 10.6 on page 469 is
The solution to the equation $T(N) = aT(N/b) + \Theta(N^k)$ is $T(N) = O(N^{log_b} a)$ if $a > b^k$
The proof includes the phrase "ignore the constant factor in $\theta(N^k)$ ..." and then goes on to use a telescoping sum to get
$T(N) = T(b^m) = a^m \sum_{i=0}^m (\frac{b^k}{a})^i$
At this point, he argues that because $a > b^k$, then the sum is a geometric series with a ratio less than 1. However, if we don't ignore the constant in the $\Theta$ expression, isn't there a constant inside the summation that would affect whether the geometric series converges?