How do you determine whether the sequence #a_n=(n+1)^n/n^(n+1)# converges, if so how do you find the limit?
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1 Answer
Mar 29, 2017
Explanation:
Note that:
#(1+1/n)^n#
is monotonically increasing with limit
So:
#lim_(n->oo) a_n = lim_(n->oo) (n+1)^n/(n^(n+1))#
#color(white)(lim_(n->oo) a_n) = lim_(n->oo) ((n+1)/n)^n/n#
#color(white)(lim_(n->oo) a_n) = lim_(n->oo) (1+1/n)^n/n#
#color(white)(lim_(n->oo) a_n) <= lim_(n->oo) e/n#
#color(white)(lim_(n->oo) a_n) = 0#
