How do you use the ratio test to test the convergence of the series #∑(4^n) /( 3^n + 1)# from n=1 to infinity?

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Answer 1 · 2016-04-05T22:45:17

Use the ratio test to find that this series diverges...

Let #a_n = 4^n/(3^n+1)#

Then:

#a_(n+1)/a_n = (4^(n+1)/(3^(n+1)+1))/(4^n/(3^n + 1))#

#=(4^(n+1)(3^n+1))/(4^n(3^(n+1)+1))#

#=4/3*(1+3^-n)/(1+3^(-n-1))#

So #lim_(n->oo) a_(n+1)/a_n = 4/3#

Since this is greater than #1#, the series diverges.