Which of the following series is/are convergent by the alternating series test?

Question

Which of the following series is/are convergent by the alternating series test?

1. 
2. 
3. 

A	I only
B	I and II only
C	I and III only
D	II and III only

Answer 1

Looking at A,B,C the answer would be B.

Looking at A although ln(n) grows more slowly than 1/x which is know to diverge, we also know the alternating series of 1/n --> (-1)^n/n does converge conditionally. In this case A would also converge conditionally.

Looking at B looks like 1/n for large values of n and based on the argument above for A would provide conditional convergence.

Looking at C looks like 2/5 in the limit of large n. Since there would be a infinite number of 2/5's and they would alternate in sign, there is no limit. Use of the ratio test shows the L value is 1 which is no determination.