Questions: Determine whether the following series converges. Justify your answer. sum from k=1 to infinity of (4+cos 2k) / k^4 Because (4+cos 2k) / k^4 ≤ 5 / k^4 and, for any positive integer k, sum from k=1 to infinity of 5 / k^4 converges, the given series converges by the Comparison Test.

Determine whether the following series converges. Justify your answer.

sum from k=1 to infinity of (4+cos 2k) / k^4

Because (4+cos 2k) / k^4 ≤ 5 / k^4 and, for any positive integer k, sum from k=1 to infinity of 5 / k^4 converges, the given series converges by the Comparison Test.

Transcript text: Determine whether the following series converges. Justify your answer. \[ \sum_{k=1}^{\infty} \frac{4+\cos 2 k}{k^{4}} \] Because $\frac{4+\cos 2 k}{k^{4}} \leq \frac{5}{k^{4}}$ and, for any positive integer $k, \sum_{k=1}^{\infty} \frac{5}{k^{4}}$ converges, the given series converges by the Comparison Test.

Solution

Solution Steps

To determine whether the series $\sum_{k=1}^{\infty} \frac{4+\cos 2 k}{k^{4}}$ converges, we can use the Comparison Test. The series $\sum_{k=1}^{\infty} \frac{1}{k^4}$ is a p-series with $p = 4$, which is known to converge because $p > 1$. We compare the given series to this p-series. Since $\frac{4+\cos 2 k}{k^4} \leq \frac{5}{k^4}$ for all $k$, and $\sum_{k=1}^{\infty} \frac{5}{k^4}$ converges, the given series also converges by the Comparison Test.

Step 1: Define the Series

We are given the series

\[ \sum_{k=1}^{\infty} \frac{4+\cos(2k)}{k^4}. \]

Step 2: Identify the Comparison Series

We compare the given series to the p-series

\[ \sum_{k=1}^{\infty} \frac{1}{k^p} \]

where $p = 4$. Since $p > 1$, we know that this p-series converges.

Step 3: Establish the Comparison

We observe that

\[ \frac{4+\cos(2k)}{k^4} \leq \frac{5}{k^4} \]

for all $k$ because $\cos(2k)$ oscillates between $-1$ and $1$. Thus, we can conclude that

\[ \sum_{k=1}^{\infty} \frac{5}{k^4} \]