Question

Use the properties of geometric series to find the sum of the series. for what values of the variable does the series converge to this sum?

4 - 8z - 16z² - 32z³ + .......

Answer 1

The sum of the series is 4 / (1 - (-2z)). The series converges to this sum for values of z that satisfy |z| < 1/2.

Step-by-step explanation:

This series is a geometric series with a common ratio of -2z. The first term of the series is 4. In a geometric series, the sum is given by the formula:

sum = a / (1 - r)

where 'a' is the first term and 'r' is the common ratio.

So, for this series, the sum is:

sum = 4 / (1 - (-2z))

To find the values of 'z' for which the series converges to this sum, we need the absolute value of the common ratio to be less than 1. So, we have:

|-2z| < 1

Simplifying, we get:

|z| < 1/2

Therefore, the series converges to the given sum for values of 'z' that satisfy the inequality |z| < 1/2.