Question in summation

$$\hbox {Is }\;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j} \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}B_{i,j}\;\hbox{?}$$

$$\hbox {Is }\;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}\;\hbox{?}$$

I think it is because even though the right side has two summation of ##i## , but both increment at the same time. So is ##j##. therefore the result is the same.

Last edited:

DrClaude
Mentor
$$\hbox {Is }\;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j} \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}B_{i,j}\;\hbox{?}$$
Is ##a_1 b_1 + a_2 b_2 = (a_1 +a_2) (b_1 + b_2)##?

$$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j}B_{i,j}$$
This is not correct. You cannot have two summations over the same index.

Thanks for the reply, but if you keep j=1, i=1,2,3......

$$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j} \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}B_{i,j}=A_{1,1}B_{1,1}+A_{2,1}B_{2,1}+A_{3,1}B_{3,1}+.......$$

$$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}\;=A_{1,1}B_{1,1}+A_{2,1}B_{2,1}+A_{3,1}B_{3,1}+......$$
If you start increment j, the series just repeat with j=2,3,4......

Because ##i## increment all at the same time. There should be no difference. This is not like trying to make

##a_1 b_1 + a_2 b_2 = (a_1 +a_2) (b_1 + b_2)##

DrClaude
Mentor
$$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j} \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}B_{i,j}$$
The way I read this is
$$\left( \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j} \right) \left( \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}B_{i,j} \right)$$
which means that addition and multiplication have been inverted, which is not correct.

$$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}\;=A_{1,1}B_{1,1}+A_{2,1}B_{2,1}+A_{3,1}B_{3,1}+......$$
If you start increment j, the series just repeat with j=2,3,4......
Because ##i## increment all at the same time.

Again, you cannot have two summations with the same index. It doens't make sense.

Ray Vickson
Homework Helper
Dearly Missed
Thanks for the reply, but if you keep j=1, i=1,2,3......

$$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j} \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}B_{i,j}=A_{1,1}B_{1,1}+A_{2,1}B_{2,1}+A_{3,1}B_{3,1}+.......$$

$$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}\;=A_{1,1}B_{1,1}+A_{2,1}B_{2,1}+A_{3,1}B_{3,1}+......$$
If you start increment j, the series just repeat with j=2,3,4......

Because ##i## increment all at the same time. There should be no difference. This is not like trying to make

##a_1 b_1 + a_2 b_2 = (a_1 +a_2) (b_1 + b_2)##

No, you are wrong: it is exactly like that. In fact, if you take ##A_{11} = a_1, A_{12}= a_2, B_{11} = b_1, B_{12} = b_2## and all other ##A_{ij}, B_{ij} = 0,## then you are claiming that ##\sum_i\sum_j A_{ij}B_{ij} = a_1 b_1 + a_2 b_2## equals ##\sum_i \sum_j A_{ij} \sum_l \sum_m B_{lm} = (a_1+a_2)(b_1+b_2),## and that is false.

No, you are wrong: it is exactly like that. In fact, if you take ##A_{11} = a_1, A_{12}= a_2, B_{11} = b_1, B_{12} = b_2## and all other ##A_{ij}, B_{ij} = 0,## then you are claiming that ##\sum_i\sum_j A_{ij}B_{ij} = a_1 b_1 + a_2 b_2## equals ##\sum_i \sum_j A_{ij} \sum_l \sum_m B_{lm} = (a_1+a_2)(b_1+b_2),## and that is false.

I really don't get this, you are using ##\sum_i \sum_j A_{ij}## on the first and ##\sum_l \sum_m## on the second one. I am using ##\sum_i \sum_j ## for both. The two sum cannot be independently incremented. When ##i## incremented by one, both has to be incremented by 1. Using what you say that ##i=1## and ##j##=1,2 only all other are zeros.

$$\hbox{For }i=1,j=1,\;\sum_i \sum_j A_{ij} \sum_i \sum_j B_{ij}=\left(\sum_i \sum_j A_{ij}\right) \left(\sum_i \sum_j B_{ij}\right)=A_{1}B_{1}$$

$$\hbox{For }i=1,j=2,\;\sum_i \sum_j A_{ij} \sum_i \sum_j B_{ij}=\left(\sum_i \sum_j A_{ij}\right) \left(\sum_i \sum_j B_{ij}\right)=A_{2}B_{2}$$

So the sum will be ##a_1b_1+a_2b_2##

Last edited:
verty
Homework Helper
I really don't get this, you are using ##\sum_i \sum_j A_{ij}## on the first and ##\sum_l \sum_m## on the second one. I am using ##\sum_i \sum_j ## for both. The two sum cannot be independently incremented. When ##i## incremented by one, both has to be incremented by 1. Using what you say that ##i=1## and ##j##=1,2 only all other are zeros.

$$\hbox{For }i=1,j=1,\;\sum_i \sum_j A_{ij} \sum_i \sum_j B_{ij}=\left(\sum_i \sum_j A_{ij}\right) \left(\sum_i \sum_j B_{ij}\right)=A_{1}B_{1}$$

$$\hbox{For }i=1,j=2,\;\sum_i \sum_j A_{ij} \sum_i \sum_j B_{ij}=\left(\sum_i \sum_j A_{ij}\right) \left(\sum_i \sum_j B_{ij}\right)=A_{2}B_{2}$$

So the sum will be ##a_1b_1+a_2b_2##

It doesn't work that way, you could even have this:

##\sum\limits_{i=0}^\infty \sum\limits_{i=0}^i A_i##

which would mean, A_0 + (A_0 + A_1) + (A_0 + A_1 + A_2) + ...

My point, each sigma has its own variable that it creates just for that summation.

thanks everybody. So I cannot count on ##i## and ##j## all increment at the same time. I have to treat is as if they are independent between the two summation?

Thanks

This is the link of the real question about summation. I just don't know how the book can move the summation inside: