(Changing) Limits of a Summation

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The discussion revolves around the interpretation of the summation notation ##\sum_{i=k}^m a_i##, clarifying that it represents the sum of the values ##a_i## from index ##k## to index ##m##. Participants emphasize the importance of correctly defining the indices and the summation symbol ##\sum##. A transformation of the summation using a different index, such as ##j##, is suggested to aid understanding. The conversation highlights the need for precise notation and consistent variable usage in mathematical expressions. Overall, the thread focuses on accurately defining and manipulating summation limits.
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Homework Statement
Good afternoon 😃 Can someone explain to me how to change the bounds of summation using this example please?
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What does ##\sum_{i=k}^m a_i## stand for?
 
Try ##j = i - m + p##.
 
fresh_42 said:
What does ##\sum_{i=k}^m a_i## stand for?
i stands for the positive integer , ai stands for a unique number and m and n are positive integers with m being greater or equal to n.
 
martina1075 said:
i stands for the positive integer , ai stands for a unique number and m and n are positive integers with m being greater or equal to n.
No, that's what the "numbers" mean. But what is the abbreviation ##\sum## for?
 
Write both sides out with the definition of the summation symbol. Shouldn't be too hard to see thay are the same.
 
martina1075 said:
fresh_42 said:
What does ##\sum_{i=k}^m a_i## stand for?
i stands for the positive integer , ai stands for a unique number and m and n are positive integers with m being greater or equal to n.
(Notice that fresh has changed some of the variables from those in the problem statement.)

Think of what @fresh_42 was getting at more like:
##\displaystyle \sum_{i=m}^n a_i## stands for:
##a_{m} + a_{m+1} + a_{m+2} + ... + a_{n-1} + a_{n} ##​

In words, sum the ##a_i ## values, where ##i## takes on values from ##m## through ##n##.

With this in mind, do a similar translation for the right hand side. It may help to use an index other than ##i##, initially.

##\displaystyle \sum_{j=p}^{p+n-m} a_{j+m-p}##

Plug ##p## in for ##j## to get the index for the first term in the sum, then plug ##p+n-m## in for ##j## to get the index for the final term.

(Added in Edit: Fixed summation per @archaic's comment in the next post.)
 
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SammyS said:
(Notice that fresh has changed some of the variables from those in the problem statement.)

Think of what @fresh_42 was getting at more like:
##\displaystyle \sum_{i=m}^n a_i## stands for:
##a_{m} + a_{m+1} + a_{m+2} + ... + a_{n-1} + a_{n} ##​

In words, sum the ##a_i ## values, where ##i## takes on values from ##m## through ##n##.

With this in mind, do a similar translation for the right hand side. It may help to use an index other than ##i##, initially.

##\displaystyle \sum_{i=p}^{p+n-m} a_{j+m-p}##

Plug ##p## in for ##j## to get the index for the first term in the sum, then plug ##p+n-m## in for ##j## to get the index for the final term.
You forgot to change ##i## to ##j## in the second symbol :)
 
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