Changing variable in a sumation

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Homework Help Overview

The discussion revolves around a manipulation of a summation involving a change of variables. The original poster is examining the transformation of the summation from one variable to another, specifically from \( k \) to \( m \), and is questioning the implications of this change on the limits of the summation.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand how the limits of the summation change when substituting variables. Some participants clarify the relationship between the original and new limits based on the transformation.

Discussion Status

The discussion has progressed with some participants providing clarifications regarding the limits of the summation after the variable change. The original poster acknowledges a resolution to their question, indicating that the discussion has moved towards understanding the reasoning behind the manipulation.

Contextual Notes

There is a focus on the implications of changing the sign of the variable in the summation and how it affects the limits, which suggests an exploration of the properties of summations and variable transformations.

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

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