Double summation: inner index = function of outer index

• MHB
• hitanshu_sachania
In summary, the conversation discusses a double summation with an inner index function of the outer index. The problem is that the relationship between the inner and outer indices, represented by M(x), is difficult to define. However, it is clear that as the value of x varies, the value of y will also vary and thus affect the inner summation.
hitanshu_sachania

Here N, a, and b are integer constants. M is also an integer but changes for every value of x, which makes the index of the second summation dependent on the first. The problem is the relationship M(x) is analytically difficult to define. Is there a way to solve/simplify this expression?

Reeii Education said:
There must be a relation between y and x according to[ which as the value of x varies, y will vary, so would M(x).
No, the only "relation" between y and x is the stated one- that y goes from 1 to M(x). For example,
$\sum_{x= 1}^3\sum_{y= 1}^{x+ 1} F(x, y)$ where "M(x)" is "x+ 1".

For x= 1 y goes from 1 to 2- the inner sum is F(1, 1)+ F(1, 2).
For x= 2 y goes from 1 to 3- the inner sum is F(2, 1)+ F(2, 2)+ F(2, 3).
For x= 3 y goes from 1 to 4- the inner sum is F(3, 1)+ F(3, 2)+ F(3, 3)+ F(3, 4).
$\sum_{x= 1}^3\sum_{y= 1}^{x+ 1} F(x, y)$= F(1, 1)+ F(1, 2)+ F(2, 1)+ F(2, 2)+ F(2, 3)+ F(3, 1)+ F(3, 2)+ F(3, 3)+ F(3, 4).

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1. What is a double summation?

A double summation is a type of mathematical operation where two summation signs are used to represent a sum of sums. It is also known as a nested summation or a double series.

2. What does the inner index represent in a double summation?

The inner index in a double summation represents the variable or index that is being summed over in the inner sum. It is usually a function of the outer index.

3. How is the inner index related to the outer index in a double summation?

The inner index is usually a function of the outer index in a double summation. This means that the value of the inner index changes for each value of the outer index.

4. What is the purpose of using a double summation?

A double summation is used to simplify complex mathematical expressions that involve nested sums. It is also used to represent the sum of a two-dimensional array of numbers.

5. Can the order of the summation signs be changed in a double summation?

Yes, the order of the summation signs can be changed in a double summation. This is known as changing the order of summation and it is a useful technique in simplifying the expression or evaluating the summation more efficiently.

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