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

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Answer 1 · 2018-07-31T13:05:17

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

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#