(Changing) Limits of a Summation

In summary, ##\sum_{i=k}^m a_i## stands for summing unique numbers ##a_i## where ##i## ranges from ##k## to ##m##, while ##j = i - m + p## represents the same sum using a different index.
  • #1
martina1075
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Homework Statement
Good afternoon 😃 Can someone explain to me how to change the bounds of summation using this example please?
Relevant Equations
As per picture
538F1044-AFD4-470C-A39F-186AF0EF7298.jpeg
 
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  • #3
Try ##j = i - m + p##.
 
  • #4
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.
 
  • #5
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?
 
  • #6
Write both sides out with the definition of the summation symbol. Shouldn't be too hard to see thay are the same.
 
  • #7
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.)
 
Last edited:
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Likes jim mcnamara and archaic
  • #8
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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Likes SammyS

Related to (Changing) Limits of a Summation

1. What is a summation?

A summation is a mathematical operation that involves adding together a sequence of numbers. It is represented by the symbol "Σ" and is commonly used in various fields of mathematics and science to express the total of a set of values.

2. What are the limits of a summation?

The limits of a summation refer to the starting and ending points of the sequence of numbers being added together. These limits are typically denoted by the numbers below and above the summation symbol, respectively. For example, in Σ n=1 to 5, the limits are 1 and 5.

3. How can the limits of a summation be changed?

The limits of a summation can be changed by altering the numbers below and above the summation symbol. This can be done by adjusting the starting and ending points of the sequence of numbers, as well as the expression being summed.

4. What is the purpose of changing the limits of a summation?

Changing the limits of a summation allows for more flexibility in performing calculations and solving problems. It can help simplify complex equations and make them easier to evaluate. It also allows for the summation of different portions of a sequence, rather than the entire sequence, which can be useful in certain applications.

5. What are some common applications of changing the limits of a summation?

Changing the limits of a summation is commonly used in various fields of mathematics and science, such as calculus, statistics, and physics. It is also used in computer science and engineering to solve problems involving large data sets and sequences. Additionally, it is used in finance and economics to calculate sums of money over time.

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