Question

How do summations work?

Answer 1

I think you're asking how the notation works, but I will also cover some examples of adding with it.

The basic structure of it is:

`sum_(n=a)^(N) f(x_1, x_2, . . ., x_m ; n)`

• The `n`, or whatever letter you want, is an index, and it indicates what spot we are in for the sum. What `n` is written to be equal to is the first index in the sum (whatever `a` is).
• The `N` in general indicates the upper limit to the sum. The sum could be written to go towards `oo`, or it could stop at the finite number `N`, depending on context.
• `f(x_1, x_2, . . . , x_m; n)` is a function of some number of variables indicated by the `x_i` (`i = 1, 2, . . . , m`), and usually is also a function of the index `n`.

For example, if we write

`sum_(n=0)^(oo) x^n`,

we are writing a condensed form of:

`x^0 + x^1 + x^2 + cdot cdot cdot`

`= 1 + x + x^2 + cdot cdot cdot`

Or, if we combine two sums, like writing

`sum_(n=0)^(oo) x^n + sum_(n=2)^(oo) (2x)^n`,

sums are commutative and their indices can be linked together, so this can also be written as:

`= sum_(n=0)^(1) x^n + sum_(n=2)^(oo) x^n + sum_(n=2)^(oo) (2x)^n`

`= 1 + x + sum_(n=2)^(oo) [x^n + (2x)^n]`

Lastly, a more explicit example would be:

`color(blue)(sum_(n=0)^(5) (2^n + 2n))`

`= [2^0 + 2(0)] + [2^1 + 2(1)] + [2^2 + 2(2)] + [2^3 + 2(3)] + [2^4 + 2(4)] + [2^5 + 2(5)]`

`= sum_(n=0)^(5) 2^n + sum_(n=0)^(5) 2n`

`= [2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5] + [2(0) + 2(1) + 2(2) + 2(3) + 2(4) + 2(5)]`

`= [1 + 2 + 4 + 8 + 16 + 32] + [0 + 2 + 4 + 6 + 8 + 10]`

`= 1 + 2 + 4 + 8 + 16 + 32 + 0 + 2 + 4 + 6 + 8 + 10`

`= color(blue)(93)`