How do you test the alternating series #Sigma (-1)^nsqrtn/(n+1)# from n is #[1,oo)# for convergence?
is an alternating series, so we can test its convergence using Leibniz's theorem, which states that an alternating series
is convergent if:
#lim_(n->oo) a_n = 0#
#a_(n+1) <= a_n#
in our case:
so the first condition is satisfied.
For the second we analyze the function:
and calculate the derivative:
we can see that
and also the second condition is satisfied.