Question

Use the ratio test or the root test to determine the convergence or divergence of the series:

1/(ln(5))^5, 1/(ln(6))^6, 1/(ln(7))^7, 1/(ln(8))^8.
a) Converges
b) Diverges

Answer 1

To determine the convergence or divergence of the series, we can use the ratio test or the root test. In this case, using the ratio test, we find that the series converges.

Step-by-step explanation:

To determine the convergence or divergence of the series, we can use the ratio test or the root test.

Ratio Test:We take the limit as n approaches infinity of the absolute value of (a(n+1)/a(n)).If this limit is less than 1, the series converges. If it is greater than 1, the series diverges.Root Test:We take the limit as n approaches infinity of the nth root of |a(n)|.If this limit is less than 1, the series converges. If it is greater than 1, the series diverges.

Let's apply the ratio test:The term in the series is 1/((ln(n))^n).The limit of (a(n+1)/a(n)) as n approaches infinity is [(ln(n+1))^n+1 / (ln(n))^n].Since it is difficult to determine the limit algebraically, we can use the fact that ln(n) grows slower than n, which implies that ln(n+1) grows slower than n+1.Therefore, the limit of (a(n+1)/a(n)) is less than 1.By the ratio test, the series converges.