None of the terms of the sequence is 0, but the limit of the sequence is. Sometimes the limit of a sequence is one of the terms. For example,

1, 1, 1, …

certainly tends to 1 and so does

1, 1/2, 1, 2/3, 1, 3/4, 1, …

But it is by no means necessary that this be so.

Perhaps a look at the formal definition of a limit will help. For a sequence of real numbers, we say that a[sub]n[/sub] tends to the limit l if:

Given any real [symbol]e[/symbol]>0, we can choose a natural number N such that |a[sub]n[/sub] - l| < [symbol]e[/symbol] whenever n>N.

In other words, if you give me any positive margin of error, I guarantee that all the terms of the sequence from some point on will approximate the limit number l to within that margin of error.

Let us apply this to the two sequences we have been discussing. I have claimed that

1, 1/2, 1/3, 1/4, … converges to 0

and that

0.9, 0.99, 0.999, 0.9999, … converges to 1.

As an example take [symbol]e[/symbol], the magin of error, to be 0.001

For the first sequence, any term after the 1000th is less than 0.001 and so is within 0.001 of the proposed limit.

For the second, every term after the 3rd is within 0.001 of 1.

If instead we take [symbol]e[/symbol] = 0.000001, then we just have to go further along the two sequences: the 1,000,000th term will do for the first sequence; the 6th term for the second.

In general, whatever positive [symbol]e[/symbol] you choose, I can guarantee that all terms from some point on will be within [symbol]e[/symbol] of my claimed limit. That is what we mean by convergence, and in particular by the value of an infinite sum.