How do you use the Root Test on the series sum_(n=1)^oo((5n-3n^3)/(7n^3+2))^n ?

1 Answer
Oct 2, 2014

Let

a_n=({5n-3n^3}/{7n^3+2})^n.

By Root Test,

lim_{n to infty}rootn{|a_n|}=lim_{n to infty}rootn{|({5n-3n^3}/{7n^3+2})^n|}

by cancelling the nth-root and the nth-power,

=lim_{n to infty}|{5n-3n^3}/{7n^3+2}|

by dividing the numerator and the denominator by n^3,

=lim_{n to infty}|{5/n^2-3}/{7-2/n^3}|=|{0-3}/{7-0}|=3/7<1

Hence, the series converges absolutely.