Question

(1 point) 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?

7−14z+28z2−56z3+⋯
sum =
domain =
(Give your domain as an interval or comma separated list of intervals; for example, to enter the region x<−1 and 2

Answer 1

The sum of the series is 7 / (1 - (-2z)), and the domain for which the series converges is (-1/2, 1/2).

The given series is a geometric series with the first term, a = 7, and the common ratio, r = -2z.

To find the sum of the series, we can use the formula for the sum of an infinite geometric series:

sum = a / (1 - r)

sum = 7 / (1 - (-2z))

To find the domain for which the series converges, we need the absolute value of the common ratio to be less than 1:

| -2z | < 1

-1 < 2z < 1

-1/2 < z < 1/2

So, the sum of the series is 7 / (1 - (-2z)), and the domain for which the series converges is (-1/2, 1/2).