Is sin(π/2*n^1/n) converges or diverges ???? Help asap , thanks

1 Answer
Jul 31, 2018

lim_(n->oo) sin(pi/2 n^(1/n)) = 1

Explanation:

Consider the sequence:

n^(1/n) = (e^lnn)^(1/n) = e^(lnn/n)

As:

lim_(n->oo) lnn/n = 0

and as e^x is continuous for x = 0:

lim_(n->oo) n^(1/n) = lim_(n->oo) e^(lnn/n) = e^((lim_(n->oo) lnn/n)) = e^0 =1

Similarly, as sinx is continuous in x=pi/2:

lim_(n->oo) sin(pi/2 n^(1/n)) = sin(pi/2 *lim_(n->oo) n^(1/n)) = sin(pi/2) = 1