Register to reply

Summation proof.

by -Dragoon-
Tags: proof, summation
Share this thread:
-Dragoon-
#1
Sep19-11, 11:06 PM
P: 292
1. The problem statement, all variables and given/known data
Show that the summation notation satisfies the following property:
[itex]\displaystyle\sum\limits_{i=1}^n(\displaystyle\sum\limits_{j=1}^m aij) = \displaystyle\sum\limits_{j=1}^m(\displaystyle\sum\limits_{i=1}^n aij) [/itex]

2. Relevant equations
N/A


3. The attempt at a solution
[itex]\displaystyle\sum\limits_{i=1}^n(\displaystyle\sum\limits_{j=1}^m aij) = \displaystyle\sum\limits_{i=1}^n ai_{1} + \displaystyle\sum\limits_{i=1}^n ai_{2} + ... +\displaystyle\sum\limits_{i=1}^n ai_{n} = \displaystyle\sum\limits_{j=1}^m(\displaystyle\sum\limits_{i=1}^n aij) [/itex]

Have I proven this sufficiently or have I skipped a step? If I skipped a step, which one was it? Thanks in advance.
Phys.Org News Partner Science news on Phys.org
FIXD tells car drivers via smartphone what is wrong
Team pioneers strategy for creating new materials
Team defines new biodiversity metric
Fredrik
#2
Sep19-11, 11:25 PM
Emeritus
Sci Advisor
PF Gold
Fredrik's Avatar
P: 9,403
1. The problem statement, all variables and given/known data
Show that the summation notation satisfies the following property: [tex]\sum_{i=1}^n\bigg(\sum_{j=1}^m a_{ij}\bigg) = \sum_{j=1}^m\bigg(\sum_{i=1}^n a_{ij}\bigg) [/tex]

2. Relevant equations
N/A


3. The attempt at a solution
[tex]\sum_{i=1}^n\bigg(\sum_{j=1}^m a_{ij}\bigg) = \sum_{i=1}^n a_{i1} + \sum\limits_{i=1}^n a_{i2} + \cdots +\sum_{i=1}^n a_{im} = \sum_{j=1}^m\bigg(\sum_{i=1}^n a_{ij}\bigg) [/tex]



I would at least have written out the step [tex]\sum_{i=1}^n(a_{i1}+\cdots+a_{im})=\sum_{i=1}^n a_{i1} + \sum\limits_{i=1}^n a_{i2} + \cdots +\sum_{i=1}^n a_{im}.[/tex] If you want to do these things rigorously, you need to avoid the ... notation and use induction.

If you use tex tags instead of itex, you don't need to type "displaystyle" all the time. (Use tex tags only when you want the math to appear on a separate line). Hit the quote button to see how I prefer to type the math above.
-Dragoon-
#3
Sep20-11, 06:13 PM
P: 292
Quote Quote by Fredrik View Post
I would at least have written out the step [tex]\sum_{i=1}^n(a_{i1}+\cdots+a_{im})=\sum_{i=1}^n a_{i1} + \sum\limits_{i=1}^n a_{i2} + \cdots +\sum_{i=1}^n a_{im}.[/tex] If you want to do these things rigorously, you need to avoid the ... notation and use induction.

If you use tex tags instead of itex, you don't need to type "displaystyle" all the time. (Use tex tags only when you want the math to appear on a separate line). Hit the quote button to see how I prefer to type the math above.
Thank you for the help and the tex tips, Fredrik.


Register to reply

Related Discussions
Summation Proof Help Calculus & Beyond Homework 15
Summation proof General Math 2
Summation proof Calculus & Beyond Homework 3
Proof of the summation formula Calculus 1
Summation proof Introductory Physics Homework 4