Double sum (Sigma) problem

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  • #1
mexion
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Homework Statement
[itex]\sum_{k=1}^{3} \sum_{j=0}^{4}k^{j}[/itex]

The attempt at a solution
so this above means:
[itex](1+1^{1}+1^{2}+1^{3}+1^{4})*(1+2^{1}+2^{2}...)*(1+3...)*...[/itex]
or
[itex](1*1^{1}*1^{2}*1^{3}*1^{4})+(1*2^{1}*2^{2}...)+(1*3...)+...[/itex]
 
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  • #2
It doesn't make a difference which order you do it in. You can see if you type the following into matlab:

syms k
syms j

symsum(symsum(k^j,j,0,4),k,1,3)

or,

symsum(symsum(k^j,k,1,3),j,0,4)

the answer is 157 either way.
 
  • #3
so 1 more question
[itex]\sum_{k=0}^{4} \sum_{j=1}^{5} (3^{k} + jk)[/itex]
how to calculate this without Matlab.
Should I transform it? (but have no idea how)
or just have to put [0;4] for k, and [1;5] for j, and add everything ?
 
  • #4
You don't need to do a transform, if you start with the sum over j you will get

\sum_{k=0}^{4} (3^k+1k+ ... 3^k+5k) = 0+ ...+ 3^4+20
 
  • #5
mexion said:
Homework Statement
[itex]\sum_{k=1}^{3} \sum_{j=0}^{4}k^{j}[/itex]

The attempt at a solution
so this above means:
[itex](1+1^{1}+1^{2}+1^{3}+1^{4})*(1+2^{1}+2^{2}...)*(1+3...)*...[/itex]
or
[itex](1*1^{1}*1^{2}*1^{3}*1^{4})+(1*2^{1}*2^{2}...)+(1*3...)+...[/itex]

It means the second one (which = 157); the first would be product_{k=1..3} sum_{j=0..4} k^j = 18755.

RGV
 

1. What is a double sum (Sigma) problem?

A double sum problem is a type of mathematical problem that involves adding together two summations, each with its own variable and range of values. The result is a sum of sums, or a double sum.

2. How do you solve a double sum problem?

The first step in solving a double sum problem is to simplify the expression by expanding the summations and combining like terms. Then, the inner sum can be evaluated using the outer sum variable as a constant. The final result is a single sum expression that can be solved using standard summation techniques.

3. What is the difference between a double sum and a single sum?

A single sum involves adding a series of terms with a single variable and a fixed range of values. A double sum, on the other hand, involves adding two summations, each with its own variable and range of values. This results in a sum of sums, or a double sum.

4. What are some common applications of double sum problems?

Double sum problems are commonly used in various fields of science, such as physics, engineering, and statistics. They are also used in computer science for algorithm analysis and in economics for complex financial calculations.

5. Are there any strategies for simplifying double sum problems?

Yes, there are several strategies for simplifying double sum problems. These include using algebraic identities, factoring, and changing the order of summations. It is also helpful to understand the properties of summations, such as the distributive and associative properties, to simplify the expression.

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