Answer
a) First four terms: $1,3,6,10$
b) Limit doesn't exist
Work Step by Step
$$\sum_{k=1}^{\infty} k$$
Part A)
$a_1 = 1$
$a_2 = 1 + 2 = 3$
$a_3 = 3 + 3 = 6$
$a_4 = 6 + 4 = 10$
Part B)
Limit doesn't exist because sum seems to be increasing without a limit as $k$ increases.