Changing variable in a sumation

[wildman]

Homework Statement

I have been shown this manipulation of a summation and I am wondering why the person could do it:
$$\sum_{k=n+1}^{0} -k 3^k$$

Now change the variables with m=-k and we get:

$$\sum_{m=0}^{-n-1} m (\frac{1}{3})^m$$

The Attempt at a Solution

I can see where the $$m (\frac{1}{3})^m$$ came from (just stick -m in for the k), but how did the summation change?

 

[cristo]

Well, k=n+1 corresponds to m=-k=-n-1, and k=0 corresponds to m=0, so the new limits are m=0, m=-n-1.

 

[wildman]

That much is obvious, but why is the 0 on the bottom and the -n-1 on the top now?

 

[wildman]

Never mind, that question is obvious also. Of course if you change the sign you would have to move the values on the summation... Problem solved.