Determine if the series converges absolutely, converges, or diverges.
-$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \ln \left[ \frac { 6 n + 2 } { 6 n + 1 } \right]$$
A) Diverges
B) converges absolutely
C) Converges conditionally

Question

Quizplus · Accepted Answer

To determine the convergence of the series, we first check for absolute convergence by considering the absolute value of the terms: $$\sum _ { n = 1 } ^ { \infty } \left| ( - 1 ) ^ { n } \ln \left[ \frac { 6 n + 2 } { 6 n + 1 } \right] \right| = \sum _ { n = 1 } ^ { \infty } \ln \left[ \frac { 6 n + 2 } { 6 n + 1 } \right]$$The terms inside the logarithm approach 1 as $n$ approaches infinity, making the logarithm approach 0. However, to determine if the series converges absolutely, we need to apply a convergence test such as the comparison test or the limit comparison test. The terms of the series decrease and approach 0, but without a specific test, it's not immediately clear if it converges absolutely.For conditional convergence, we apply the alternating series test (Leibniz test), which requires that the absolute value of the terms decreases monotonically to 0. The function $\ln \left[ \frac { 6 n + 2 } { 6 n + 1 } \right]$ does decrease monotonically to 0 as $n$ increases. Therefore, the series passes the alternating series test and converges conditionally. The series does not converge absolutely because the terms of the series, when taken in absolute value, form a series that does not meet the criteria for convergence of a p-series or comparison with a known convergent series directly. The behavior of the logarithmic function in this context does not simplify to a form that allows for easy application of these tests, but the key insight is that the decrease to 0 is not sufficient to ensure absolute convergence without further analysis.

Determine If the Series Converges Absolutely, Converges, or Diverges ∑n=1∞(−1)nln⁡[6n+26n+1]\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \ln \left[ \frac { 6 n + 2 } { 6 n + 1 } \right]n=1∑∞​(−1)nln[6n+16n+2​]

Determine If the Series Converges Absolutely, Converges, or Diverges $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \ln \left[ \frac { 6 n + 2 } { 6 n + 1 } \right]$