Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \ln n } ?$
A)Its sequence of partial sums is not bounded from above.
B)Its sequence of partial sums is decreasing.
C)It is absolutely convergent.
D)It is conditionally convergent.
E)It is divergent.

Question

Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \ln n } ?$ 
A)Its sequence of partial sums is not bounded from above.
B)Its sequence of partial sums is decreasing.
C)It is absolutely convergent.
D)It is conditionally convergent.
E)It is divergent.

Quizplus · Accepted Answer

This series is an example of an alternating series, where the terms alternate in sign. It is conditionally convergent because it satisfies the conditions of the Alternating Series Test: the absolute values of the terms decrease to 0 as $n$ approaches infinity ($\lim_{n \to \infty} \frac{1}{\ln(n)} = 0$), and the terms alternate in sign. However, it is not absolutely convergent because the series of absolute values $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \ln n }$ diverges, as it can be compared to the harmonic series or tested with the integral test, showing that the integral of $1/\ln(x)$ from 2 to infinity diverges.

Which of the Following Is True for the Infinite Series ∑n=1∞(−1)nln⁡n?\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \ln n } ?∑n=1∞​lnn(−1)n​?

Which of the Following Is True for the Infinite Series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \ln n } ?$