Question

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

Answer 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.

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.