How do you show that the series #1/2+2/3+3/4+...+n/(n+1)+...# diverges? Calculus Tests of Convergence / Divergence Nth Term Test for Divergence of an Infinite Series 1 Answer Cesareo R. Mar 1, 2017 Is divergent. Explanation: #1/n < n/(n+1) # and #sum_(k=1)^oo 1/k < sum_(k=1)^oo k/(k+1)# but #sum_(k=1)^oo 1/k# is the so called harmonic series which is divergent. So #sum_(k=1)^oo k/(k+1)# is also divergent. Answer link Related questions What is Nth Term Test for Divergence of an infinite series? How do you use the Nth term test for divergence on an alternating series? How do you know When to use the nth term test for divergence on an infinite series? How do you use the Nth term test on the infinite series #sum_(n=1)^ooln((2n+1)/(n+1))# ? How do you use the Nth term test on the infinite series #sum_(n=2)^oon/ln(n)# ? How do you use the Nth term test on the infinite series #sum_(n=1)^ooarctan(n)# ? How do you use the Nth term test on the infinite series #sum_(n=1)^oorootn(2)# ? How do you use the Nth term test on the infinite series #sum_(n=1)^oo(n(n+2))/(n+3)^2# ? How do you use the Nth term test on the infinite series #sum_(n=1)^ooe^2/n^3# ? How do you use the Nth term test on the infinite series #sum_(n=1)^oosin(n)# ? See all questions in Nth Term Test for Divergence of an Infinite Series Impact of this question 8429 views around the world You can reuse this answer Creative Commons License