Question

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

Answer 1

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.