Root Test — Question 4

Question 4 – Solution

Step 1: Take the $n$th root.

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

Step 2: Evaluate the polynomial-root factor.

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

Step 3: Apply the test.

$$L=\lim_{n\to\infty}\frac{n^{5/n}}4=\frac14<1.$$ 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.