Question

Determine whether the geometric series 192 + 48 + 12 + ... converges or diverges, and identify the sum if it exists.

A.) Converges: 768

B.) Diverges

C.) Converges; 64

D.) Converges; 256

Answer 1

D.) Converges; 256

Explanation:

x0= 192

x1 = 48 = 192/4

x2 = 12 = 192/(4 x 4)

Therefore, this series can be written as:

`x_n = (192)/(4^n)`

Applying limits at infinity:

`\lim_(n \to \infty) x_n= \lim_(n \to \infty) ((192)/(4^n)) = (192)/(\infty)=0`

Since the terms of the series tend to zero, we can affirm that the series converges.

The sum of an infinite converging series is:

`S=(x_0)/(1-r) \\S=(192)/(1-(1)/(4) )\\S=256`

Thus, the answer is D.) Converges; 256