Series can sometimes be confusing (or at least useful when proving that the sum of an infinite series of positive numbers is negative. or that 0 = 1), so the definition of the value of a series might be helpful. (To be pedantic, the expression "sum of a series" is redundant since a series is a sum).
A series T1 + T2 + T3 + ... = sumi=1 to infinity Ti is said to converge to a value S if the limit S = limn->infinity sumi=1 to n Ti exists. In that case the series is said to be convergent. If no limit exists, it is divergent. Note that the sum up to n can be bounded, even for a divergent series: e.g. 1-1+1-1+1...
Historically, this definition is "backwards", because series were defined first. The modern concept of limits is a later generalisation.