Final answer:
To determine if the series is convergent or divergent, use the comparison test. Compare the terms of the given series with a convergent geometric series. The given series is convergent.
Step-by-step explanation:
To determine whether the series 25/3+125/9+625/27 ... is convergent or divergent, we can use the comparison test. The comparison test states that if the terms of a series can be bounded above or below by the terms of another series that is known to be convergent or divergent, then the original series will have the same convergence behavior. Let's compare the terms of the given series with the terms of the geometric series 1/3 + 1/9 + 1/27 + ... which is known to be convergent.
Since the numerators of the given series are the powers of 5 (25, 125, 625, ...), we can rewrite the series as (5/3)^2 + (5/3)^3 + (5/3)^4 + ... . Comparing the terms with the geometric series, we can see that (5/3)^n is always greater than or equal to 1/3^n for all positive values of n. Therefore, the given series is bounded below by the convergent geometric series.
By the comparison test, we can conclude that the series 25/3 + 125/9 + 625/27 ... is convergent.