Final answer:
The series -8/5 + 32/25 - 128/125 + ... is a geometric series with a ratio of -4/5. Since the absolute value of the ratio is less than 1, the series is convergent.
Step-by-step explanation:
The series in question is -8/5 + 32/25 - 128/125 + ... . This sequence resembles a geometric series, where each term is derived from its predecessor by multiplying by a constant ratio (r). To determine whether it is convergent or divergent, we need to find the value of this ratio and apply the convergence criteria for geometric series.
Examining the series, we see that each term is the result of multiplying the previous term by -4/5. Since -1 < -4/5 < 1, this ratio falls within the range where a geometric series would converge. Therefore, the series is convergent.
The ratio test for geometric series states that if the absolute value of r is less than 1, the series converges; otherwise, it diverges. In this case, |r| = |-4/5| = 4/5, which is less than 1, confirming that the series is indeed convergent.