Final answer:
The first four terms of the series are 10, 59/20, 227/100, and 578/625. The series converges as the terms approach zero as n gets larger.
Step-by-step explanation:
The given series is ∑n=0 to[infinity] ( (7/4ⁿ) + 3/5ⁿ). To find the first four terms, substitute n = 0, 1, 2, and 3 into the series. The first term is (7/4⁰) + (3/5⁰) = 7 + 3 = 10. The second term is (7/4¹) + (3/5¹) = 7/4 + 3/5 = 59/20. The third term is (7/4²) + (3/5²) = 7/16 + 3/25 = 227/100. The fourth term is (7/4³) + (3/5³) = 7/64 + 3/125 = 578/625.
To find the sum of the series or show that it diverges, we can analyze the behavior of the individual terms as n approaches infinity. As n gets larger, the denominator of both terms (4ⁿ and 5ⁿ) will also get larger. This causes the terms to approach zero. Since the terms of the series converge to zero, the series converges. However, finding the exact sum of the infinite series is beyond the scope of this question.