Use the Ratio Test to determine whether the series is convergent or divergent.∞n4nn = 1

1. Break Down the Problem

We need to apply the Ratio Test to the series given by

$\sum_{n=1}^{\infty} \frac{4^n}{n}$

1. Define the term $a_n = \frac{4^n}{n}$.
2. Compute the ratio $\frac{a_{n+1}}{a_n}$.

2. Relevant Concepts

The Ratio Test states that for a series $\sum a_n$, we consider the limit

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

• If $L < 1$, the series converges.
• If $L > 1$ or $L = \infty$, the series diverges.
• If $L = 1$, the test is inconclusive.

3. Analysis and Detail

1. Calculate $a_{n+1}$:

$a_{n+1} = \frac{4^{n+1}}{n+1} = \frac{4 \cdot 4^n}{n+1}$

2. Calculate the ratio $\frac{a_{n+1}}{a_n}$:

$\frac{a_{n+1}}{a_n} = \frac{\frac{4 \cdot 4^n}{n+1}}{\frac{4^n}{n}} = \frac{4 \cdot 4^n \cdot n}{4^n \cdot (n+1)} = \frac{4n}{n+1}$

3. Now compute the limit as $n$ approaches infinity:

$L = \lim_{n \to \infty} \frac{4n}{n+1} = \lim_{n \to \infty} \frac{4n}{n(1 + \frac{1}{n})} = \lim_{n \to \infty} \frac{4}{1 + \frac{1}{n}} = 4$

4. Verify and Summarize

Since $L = 4 > 1$, we conclude that the series diverges.

Final Answer

The series $\sum_{n=1}^{\infty} \frac{4^n}{n}$ diverges by the Ratio Test.