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? 3/14x, 28/x², 56/x³,........

Answer 1

The sum of the series can be found using the properties of geometric series. The series converges for all values of x such that |x| > 2.

Step-by-step explanation:

The given series can be written as:

3/14x, 28/x², 56/x³, ...

To find the sum of the series, we need to check if the series converges. For a geometric series to converge, the absolute value of the common ratio, r, must be less than 1: |r| < 1.

In this series, the common ratio is 14x / 28 = 1/2x. So, to find the values of x for which the series converges, we need to solve the inequality:

|1/2x| < 1

Simplifying the inequality, we have:

1/2|x| < 1

|x| > 2

Therefore, the series converges for all values of x such that |x| > 2.