test the series for convergence

Question

Test the series below for convergence using the Ratio Test. $\sum_{n = 1}^{\infty} \frac{n^{3}}{{0.9}^{n}}$

The limit of the ratio test simplifies to $lim_{n \to \infty} | f (n) |$ where

$f (n) =$ Incorr

The limit is:

Please find f(n) and the limit!!! I would really appreciate it!

PlIncorrec

Adam B. · Accepted Answer

We could have said from the beginning that series diverges since the limit of the sequence that generate the series is different than zero.

Notice that lim a_n = lim n³/ [ 0.9]ⁿ = = lim ( 10/9 )ⁿ n³ ≠ 0. Hence the given series diverges.

On another note now applying the ratio test we have lim| ( a_n+1/ a_n)| =lim|{ (n+1)³/(0.9)ⁿ⁺¹}/{ n³/(.9)ⁿ}|=

lim{ (n+1)³/(0.9)ⁿ⁺¹}/{ n³/(.9)ⁿ}= lim { (10/9)[ (n+1)³/n³] = 10/9 >1.

Hence the series diverges

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