Summation

Our strategy for teaching Summation will start with “learn from the example”.

$\displaystyle \sum_{i=1}^3 k = k + k + k = 3k$

$\displaystyle \sum_{i=1}^3 x = 1 + 2 + 3 = 6$

$\displaystyle \sum_{i=1}^5 x = 1 + 2 + 3 + 4 + 5= 15$

$\displaystyle \sum_{i=1}^3 2x = 2 + 4 + 6 = 12$

$\displaystyle \sum_{i=1}^3 x_i = x_1 + x_2 + x_3$

What happens when we have a double summation?

$\displaystyle \sum_{i=1}^n \sum_{j=1}^m f(x,y) = \sum_{j=1}^m \sum_{i=1}^n f(x,y)$

Consider what happens with the following simple example:

$\displaystyle \sum_{i=1}^3 \sum_{j=1}^4 (i)(2j) = \begin{matrix} (1)(2) + (1)(4) + (1)(6) + (1)(8) \\ (2)(2) + (2)(4) + (2)(6) + (2)(8) \\ (3)(2) + (3)(4) + (3)(6) + (3)(8) \end{matrix} = 20 + 40 + 60 = 120$

$\displaystyle \sum_{j=1}^4 \sum_{i=1}^3 (i)(2j) = \begin{matrix} (2)(1) + (2)(2) + (2)(3) \\ (4)(1) + (4)(2) + (4)(3) \\ (6)(1) + (6)(2) + (6)(3) \\ (8)(1) + (8)(2) + (8)(3) \end{matrix} = 12 + 24 + 36 + 48 = 120$

Appendix A

Summation is associative.

If we have three summations, then we have a cube of term (we add terms), we can argue that changing the order of summations changes the locations of terms in the cube, but the same terms are present in each Cube representing a scenario oh, thus the answer to which summation will be the same.

Appendix B

$\displaystyle \sum (a_i + b_i)c_i = \sum a_ic_i + \sum b_ic_i$ $\displaystyle \sum (a_i + b_i)c_i = \sum a_ic_i + \sum b_ic_i$

Appendix C

Every term in a summation has to be the same thing and the answer to the calculation also has to be that same thing.

When the answer is a number, then every term has to be a number. This is helpful to us in the following:

$a_1 b_1 + a_2 b_2 + a_3 b_3 = c$

If we are told that c is a scalar, and then we are told that every b is a scalar, then the only way for this to be true it is for every a to be a number.

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