Consider the infinite geometric series below. a. Write the first 4 terms of the series b. Does the series diverge or converge? c. If the series has a sum, find the sum. ∞ Σ n=2 ( - 2) n-1

(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.

Step-by-step 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.