>>11153042You are correct, convergent means that it has a definite well defined numerical value. What you seem to be confused about is that both SEQUENCES and SUMS can be convergent, and these are two completely different concepts (albeit related).
A sum is of course a sum of numbers - e.g. you mention 1 + 1/2 + 1/3 + ... (note the plus). A sequence is an ordered LIST of numbers, but we're nut actually adding them - e.g. {1, 2, 3, . . . } is a very simple sequence.
Returning to your example, if you consider the values of 1/n where n is a positive integer, you have SEQUENCES like {1, 1/2, 1/3, ..., 1/n}, or you can consider sums like 1 + 1/2 + 1/3 + .... + 1/n. In which case, if we let n = 4 we have the SEQUENCE {1, 1/2, 1/3, 1/4}, whereas the sum would be 1+(1/2)+(1/3)+(1/4) = (12/12) + (6/12) + (4/12) + (3/12) = 25/12.
In other words, the sequence is the actual list of numbers itself, whereas the sum is the value to get when you add them.
Now sometimes all the numbers in a sequence get closer and closer to some value. In the case of {1, 1/2, ... , 1/n}, the value is 0, since as n get larger and larger, the fraction 1/n gets smaller & smaller. We therefore say that the sequence converges to 0.
Sums can also converge. A sum converges when it is equal to some real number value no matter when you consider the limit of the summation as n approaches infinity. For example, clearly the sum 1+2+3+...+n cannot converge since it's value approaches infinity as n approaches infinity. On the other hand if you consider sums of really, really small numbers, they often do equal a real number value. For instance, if you consider the function 1/(x^2), this function converges - according to Wikipedia, it's value is approx. 1.645.
It can get confusing with some problems like the example you gave, because it turns out the SEQUENCE of the function converges, but the SUMMATION of the function doesn't converge. Namely the sum 1+(1/2)+...+(1/n) actually tends toward infinity.
