How to Change the Order of a Double Sum?

• geor
In summary, the problem is changing the order of a double sum and taking x out of the inner sum. The solution involves using the Iverson bracket and results in a polynomial of x in the form a_n*x^n+...+a_1*x+a_0. The final formula is \sum_{j=0}^{n-1} x^j \sum_{i=j}^{n-1} a_i \binom{i}{j} b^{i-j}.
geor
[SOLVED] Changing order of a double sum

Hello everybody,

I am a bit confused here, any help would be greatly appreciated..
I have this double sum:

$$\sum_{i=0}^{n-1}a_i \sum_{j=0}^{i} {i \choose j} b^{i-j}x^j$$

How can I take x out of the inner sum?

Thank you very much in advance...

Last edited:
Would it be helpful to use the binomial theorem?

$$\sum_{j=0}^{i} {i \choose j} b^{i-j}x^j = (b + x)^i$$

Thanks for taking the time to answer!

Well, no, I started from there, I want to write this as a polynomial of x in the usual way, that is, in the form:

a_n*x^n+...+a_1*x+a_0

I want to have only x there...

It is possible, is it not?!

Whoops, I thought I had hit the submit button hours ago, but apparently I didn't.

For changing order of sums, the Iverson bracket
http://xrl.us/befjqx
is a useful tool.

$$\sum_{i=0}^{n-1}a_i \sum_{j=0}^{i} \binom{i}{j} b^{i-j} x^j$$
$$= \sum_{i,j} [0 \le j \le i][0 \le i \le n-1] a_i \binom{i}{j} b^{i-j} x^j$$
$$= \sum_{i,j} [0 \le j \le i \le n-1] a_i \binom{i}{j} b^{i-j} x^j$$
$$= \sum_{j,i} [0 \le j \le n-1][j \le i \le n-1] a_i \binom{i}{j} b^{i-j} x^j$$
$$= \sum_{j=0}^{n-1} x^j \sum_{i=j}^{n-1} a_i \binom{i}{j} b^{i-j}.$$

Thanks so much for the help!

What a nice tool! I was struggling for so much time trying to change that variables!

1. What is the purpose of changing the order of a double sum?

The purpose of changing the order of a double sum is to simplify and potentially speed up calculations. By rearranging the terms in a double sum, it may be possible to reduce the number of operations needed to calculate the sum, making the process more efficient.

2. Can the order of a double sum be changed without affecting the result?

In most cases, the order of a double sum can be changed without affecting the result. However, there are certain cases where the order of summation can lead to a different result, particularly when dealing with infinite series or non-commutative operations.

3. What are some common strategies for changing the order of a double sum?

Some common strategies for changing the order of a double sum include factoring out constants, using the distributive property, and using the associative and commutative properties of addition.

4. Are there any limitations to changing the order of a double sum?

Yes, there are some limitations to changing the order of a double sum. For example, if the terms in the double sum do not converge absolutely, changing the order of summation may lead to a different result. Additionally, certain operations, such as division, may not be associative, limiting the options for rearranging terms.

5. How does changing the order of a double sum relate to the concept of convergence?

Changing the order of a double sum can affect the convergence of the sum. For example, if the original double sum converges absolutely, changing the order of summation will not affect the convergence. However, if the original sum only converges conditionally, changing the order of summation may change the result or cause the sum to diverge.

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