Root Test — Question 4

*Question 4 Use the Root Test to determine whether $\displaystyle\sum_{n=1}^{\infty}\frac{n^5}{4^n}$ converges. Explain explicitly why the polynomial factor does not affect the final root limit.

*Question 4 – Solution

$$\sqrt[n]{\frac{n^5}{4^n}}=\frac{n^{5/n}}4.$$

Take logarithms: $$\log(n^{5/n})=\frac{5\log n}{n}\longrightarrow0.$$ Exponentiating gives $n^{5/n}\to e^0=1$.

$$L=\lim_{n\to\infty}\frac{n^{5/n}}4=\frac14<1.$$ Therefore the series converges absolutely. Under an $n$th root, any fixed power $n^p$ becomes $n^{p/n}\to1$, so the exponential denominator determines the result.