There are different kinds of series. There's harmonic, arithmetic, geometric.
Infinite arithmetic series will never have sums. This is because if the common difference is non-zero (infinite series), there's always a next number, and the series will not converge.
In the case of a geometric series, infinite kinds of these series will have sums as long as the series will converge. The series will converge if the absolute value of the common ratio is less than 1. In such case, the sum of the infinite geometric series is given by the following:
`S_infty = (a_1)/(1-r)` where `a_1` is the first term, and `|r| < 1` .
In the case of a harmonic series, it is more complicated. There are harmonic series that converge, and there are those that diverge, and techniques in calculus are used to determine convergence or divergence. An example of a harmonic series (infinite) that has a sum is the alternating series:
`sum^infty_(k=1) ((-1)^(n+1))/n` whose sum is equal to `ln(2)` .