Does the series #sum_(n=1)^oo((2^(n-1)3^(n+1))/n^n)# converge or diverge?

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Answer 1 · 2017-11-15T22:25:12

The series:

#sum_(n=1)^oo (2^(n-1)3^(n+1))/n^n#

is convergent.

Evaluate the ratio:

#abs(a_(n+1)/a_n) = (( 2^n 3^(n+2))/(n+1)^(n+1))/((2^(n-1)3^(n+1))/n^n#

#abs(a_(n+1)/a_n) = ( 2^n 3^(n+2))/ (2^(n-1)3^(n+1)) n^n / (n+1)^(n+1)#

#abs(a_(n+1)/a_n) = 6/(n+1) n^n/ (n+1)^n#

#abs(a_(n+1)/a_n) = 6/(n+1) 1/ ((n+1)/n)^n#

#abs(a_(n+1)/a_n) = 6/(n+1) 1/ (1+1/n)^n#

As:

#lim_(n->oo) 1/ (1+1/n)^n = 1/e#

we can conclude that:

#lim_(n->oo) abs(a_(n+1)/a_n) = lim_(n->oo) 6/(n+1) 1/ (1+1/n)^n = 6/e lim_(n->oo) 1/(n+1) = 0#

and then based on the ratio test the series is convergent.