Use the nth term test for divergence to determine which, if any, of the following infinite series diverge(s)

Question

Use the nth term test for divergence to determine which, if any, of the following infinite series diverge(s)

1. 
2. 
3. 

A	All of them
B	I only
C	II only
D	II and III

Answer 1

All diverge. For n >= 2, all terms of all series are greater than 1 and, in fact, greater than or equal to n. (for n = 2, sin pi/2 = 1 = 2/n and is > 2/n for all n > 2, as sin 0 = 0 and the derivative is cos is monotone decreasing on [0, pi/2], so n^2 sin pi/n >= n^2*2/n=2n; n^3 + 1 >n(n+2) = n^2 + 2n for all n >= 2, as n+2 <= n^2 for n >= 2.)

Answer 2

Since a_n does not approach 0, all of these series diverge.