Final answer:
The sequence a_n = (5n + 1) / 7^n converges to the limit of 0 as n approaches infinity.
Step-by-step explanation:
To determine whether the sequence converges or diverges, we must examine the behavior of the sequence an = (5n + 1) / 7n as n approaches infinity. Observe that as n becomes very large, the exponential term in the denominator (7n) increases much faster than the linear term in the numerator (5n). This causes the overall value of an to decrease toward zero.
The dominant term in the denominator, 7n, grows at an exponential rate, which will always outpace the linear growth of the numerator. Therefore, the sequence has a limit of 0 as n goes to infinity.
In summary, the sequence an converges to the limit of 0.