The Alternating Series Test tells us that if we have a series #sum(-1)^nb_n#, where #b_n# is a sequence of positive terms, the series converges if
a) #b_n>=b_(n+1)#, IE, the sequence is ultimately decreasing for all #n.#
b) #lim_(n->oo)b_n=0#
We should note that we don't need to have #(-1)^(n+1)# -- any term that causes alternating signs, such as #cos(npi), (-1)^(n-1), (-1)^(n+1)#, is okay.
Here, we see #b_n=n/(10n+5)#. Taking the limit,
#lim_(n->oo)n/(10n+5)=1/10 ne 0# -- the Alternating Series Test is inconclusive here.
However, take the limit of the overall sequence:
#lim_(n->oo)(-1)^(n+1)n/(10n+5) ne 0# -- we can say this because although the limit does not truly exist, we can convince ourselves that it alternates signs and gets closer to #1/10,# causing divergence by the Divergence Test.
