a_n = nsin(1/n) Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges, find its limit.

a_n=nsin(1/n)
Apply n'th term test for divergence, which states that, 
If lim_(n->oo) a_n!=0 , then sum_(n=1)^ooa_n diverges
lim_(n->oo)nsin(1/n)=lim_(n->oo)sin(1/n)/(1/n)  
Apply L'Hospital's rule,
Test L'Hospital condition:0/0
=lim_(n->oo)(d/(dn)sin(1/n))/(d/(dn)1/n) 
=lim_(n->oo)(cos(1/n)(-n^(-2)))/(-n^(-2))
=lim_(n->oo)cos(1/n)
lim_(n->oo)1/n=0
lim_(u->0)cos(u)=1 
By the limit chain rule,

=1!=0 
So, by the divergence test criteria series diverges.