The alternating series test is given as:
, where , converges if is a monotone decreasing sequence with limit zero.
I have 3 problems,I am asked to investigate these for conditional or absolute convergence, or divergence.
for problem a) :
I think i should transform (a) into the form and then prove that is positive, and a monotone decreasing sequence with limit zero, to show that (a) converges. But how exactly can i show ? Don't know how to find the limit in this form ! Also,to test for absolute convergence of the series, do i merely test the series but exclude the factor ? i.e to test for absolute convergence do i test the convergence of:
for b) :
Here r cannot be 1! do i just check from r=2? Getting (b) into the form to use the alternating series test i get
which leaves to be proven positive,and decreasing to limit zero. I thought about multiplying both numerator and denominator by n,to divide all terms by a dominant term ? to get
This seems to tend to zero as n tends to infinity. But now for absolute convergence, do i test to see if this converges ?