It is very common to encounter series for which it is difficult, or even virtually impossible, to determine the sum exactly. Often you try to evaluate the sum approximately by truncating it, i.e. having the index run only up to some finite $N\text{,}$ rather than infinity. But there is no point in doing so if the series diverges. So you like to at least know if the series converges or diverges. Furthermore you would also like to know what error is introduced when you approximate $\sum_{n=1}^\infty a_n$ by the “truncated series” $\sum_{n=1}^Na_n\text{.}$ That's called the truncation error. There are a number of “convergence tests” to help you with this.