>>13480967These are all finite sums and so make sense. Now the idea is to define the infinite sum as the number, if such exists (it might not), that the finite sums S_n approach as n gets bigger and bigger. A metric is simply the device that allows you to measure distances between numbers, and hence make sense of what it means for a sequence of numbers to approach another. Strictly speaking, if R is a number structure (for example R could be the integers Z or the rational numbers Q or the real numbers) then a metric on R is a function d which takes as arguments two numbers x,y in R and returns a real number d(x,y). It has to satisfy the following properties
1. d(x,x) = 0 for all x in R. (intuitively: the distance from any number to itself is zero)
2. d(x,y) + d(y,z) >= d(x,z) for all x,y,z in R (intuitively: the distance from x to z is always shorted than a roundabout distance first from x to y then from y to z).
3. d(x,y) > 0 for all x,y in R such that x is not y (intuitively: the distance between distinct numbers is nonzero).
The prototypical example of such a distance on the rational numbers is d(x,y)= |x-y|, the absolute value of a rational number. For example d(3,5) = d(5,3) = |3-5|=2, and d(3/2, 1) = 1/2.
When we have such a distance, called a metric, d, we say that the infinite sum a_1+a_2+.... is equal to a number L in the same structure if d(S_n, L) goes to zero as we make n larger and larger. In other words, for any e>0 there is a natural number N such that for all natural numbers n>N, d(S_n, L)< e.
If there is no such number L, then we say the sum is undefined.
