I've been given a series in summation notation. The lower limit is n=1 and the limit above the sigma is infinity. I'm asked to write the first four terms and if the series diverges or converges (and I understand those parts). But then it asks me for the to find the sum of the series. How can it have a sum if it goes on infinitely?

For
a series ∑_{n=1}^{∞}
a_{n }consider
the sequence of its partial sums s_{1}
= a_{1},
s_{2}
= a_{1}
+a_{2},
s_{3}
= a_{1}
+ a_{2}
+ a_{3},
..._{ , }

s_{n}
= ∑_{k=1}^{n}
a_{k },
… . The series converges if and only if the sequence s_{n}
converges and the sum of the series is the limit of the sequence s_{n}.

Example.
The series ∑_{n=1}^{∞}
(1/2)^{n }converges
and its sum is 1 because s_{n}
= ∑_{k=1}^{n}
(1/2)^{k}
= 1/2
+ 1/4 + 1/8 + ... +(1/2)^{n}
= 1 – (1/2)^{n}
→ 1 as n → ∞.