# Example: Changing the Summation Limits

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## Problem

Change the following summation



so that the index of summation start at 1 instead of at 3.

## Complete Solution

Although not necessary, we will use two method for solving this problem: Method 1 and Method 2.

### Method 1

Introducing the transformation



so that when  gives us



as desired.

### Method 2

Our procedure is to add and subtract terms in the sum to shift our index to 1:



as desired. Therefore



## Discussion of Some Steps

### Method 1

The transformation , was chosen to that the index  would start at 1.

### Method 2

Most steps in this approach involved straightforward algebraic manipulation. Steps (3) and (5) involve adding and subtracting terms in a way that will allow us to change the summation limits.

More precicesly, in Step (3) we added and substracted the  term, and in Step (5) we added and subtracted the  term.

## Potential Challenges

### Method 2 Requires More Work, So Why Should I Use It?

You may not have a choice. In some circumstances, you need to convert your sum in a way that does not change the general term.

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