Hence, for ,

We denote by for all . Thus,

and

The letter in (3.67) has no meaning, and can be replaced by any symbol that has no meaning in the present context. Thus .

i

even though my definition of summation is not strictly applicable here (since is not defined for all ).

There are many formulas associated with summation notation that are easily proved by induction; e.g., let be functions from to an ordered field , and let . Then

- If for all , then

Proof: (By induction.) When , (3.72) says which is true since both sides are equal to . Now suppose that (3.72) is true for some . Then

so

Hence, if (3.72) holds for some , it also holds when is replaced by . By induction (3.72) holds for all .

and

Hence

and it follows that

i.e.

Here I have derived the formula (3.72). If you write out the argument from line (3.75) to line (3.76), without using s, and using only properties of sums that we have explicitly proved or assumed, you will probably be surprised at how many implicit assumptions were made above. However all of the assumptions can be justified in a straightforward way.

Proof: Let . The formula (3.72) for a finite geometric series shows that

This formula also holds when , since then both sides of the equation are equal to zero, so

This proves our formula in the case . When , equation (3.78) says

which is true, so we will suppose that . Then by (3.80) we have

whereas if is an ordered field, then does not factor in the form , where and are in . (If for all , then by taking we would get , which is false since in any ordered field.)

- a)
- .
- b)
- . (Here .)
- c)
- .
- d)
- .
- e)
- .
- f)
- .

By looking at and using the known formula (3.72), derive the formula

(3.85) |

(3.87) |