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.
infinity
sigma
n=2
(-2)^n-1
Data unclear.
To find the first 4 terms of the given infinite geometric series, we can substitute the values of n from 2 to 5 into the formula (-2)^(n-1).
a. The first 4 terms of the series are:
(-2)^(2-1) = -2^1 = -2
(-2)^(3-1) = -2^2 = 4
(-2)^(4-1) = -2^3 = -8
(-2)^(5-1) = -2^4 = 16
b. To determine whether the series converges or diverges, we need to find the common ratio (r) of the series. In this case, the common ratio is -2. For a geometric series to converge, the absolute value of the common ratio (|r|) must be less than 1. Since |(-2)| = 2, which is greater than 1, the series does not converge. Therefore, it diverges.
c. Since the series diverges, it does not have a finite sum. The sum of a divergent series goes to infinity or negative infinity, depending on the values of the terms.