Answer:
(a) The first four terms of the series are:
-1, -2/3, -1/2, -2/5
(b) The series converges
(c) The sum does not exist.
Explanation:
Given the geometric series:
Σ(-2)n^(-1) From n = 2 to ∞
(a) Let a_n = (-2)n^(-1)
This can be rewritten as
a_n = -2/n
a_2 = -2/2 = -1
a_3 = -2/3
a_4 = -2/4 = -1/2
a_5 = -2/5
So, we have the first 4 terms of the series as
-1, -2/3, -1/2, -2/5
(b) a_n = -2/n
a_(n+1) = -2/(n+1)
|a_n/a_(n+1)| = |-2/n × -(n+1)/2|
= |-(n+1)/n|
= (n+1)/n
= 1 + 1/n
Suppose the series converges, the
Limit as n approaches infinity of
1 + 1/n exist.
Lim(1 + 1/n) as n approaches infinity
= 1 (Because 1/∞ = 0)
Therefore, the series converges.
The radius of convergence is 1.
(c) The sum does not exist.