Question

# Use the Ratio Test to determine the convergence or divergence of the series. $\sum_{n=1}^{\infty} \frac{n^{3}}{4^{n}}$

Solution

Verified
4.4 (6 ratings)
4.4 (6 ratings)
Step 1
1 of 3

\begin{align*} \lim\limits_{n \to \infty} \left| \dfrac{a_{n+1}}{a_{n}}\right| &= \lim\limits_{n \to \infty} \left| \dfrac{(n+1)^3}{4^{n+1}}\div \dfrac{n^3}{4^n} \right| \\ &= \lim\limits_{n \to \infty} \dfrac{(n+1)^3}{4n^3} \\ &= \dfrac{1}{4} \end{align*}

$a_{n} = \dfrac{n^3}{4^n}$, now we look for the limit of $a_{n+1}$/$a_{n}$.

## Recommended textbook solutions

#### Calculus

9th EditionISBN: 9780547167022 (10 more)Bruce H. Edwards, Ron Larson
12,413 solutions

#### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (6 more)James Stewart
11,083 solutions

#### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (4 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,050 solutions

#### The Practice of Statistics for the AP Exam

5th EditionISBN: 9781464108730 (2 more)Daniel S. Yates, Daren S. Starnes, David Moore, Josh Tabor
2,433 solutions