Math Snap

STEP 1

1. We are dealing with an infinite series $\sum_{n=1}^{\infty} \frac{5n^3}{n^4+7}$.
2. The goal is to determine the convergence or divergence of the series.
3. We will use the Limit Comparison Test to analyze the series.

STEP 2

1. Identify a suitable comparison series.
2. Apply the Limit Comparison Test.
3. Determine the convergence or divergence of the original series based on the test.

STEP 3

Identify a suitable comparison series. We observe that for large $n$, the term $\frac{5n^3}{n^4+7}$ behaves like $\frac{5n^3}{n^4} = \frac{5}{n}$.
Thus, we compare it with the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$, which is known to diverge.

STEP 4

Apply the Limit Comparison Test. We calculate the limit:
$$\lim_{n \to \infty} \frac{\frac{5n^3}{n^4+7}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{5n^4}{n^4+7}$$ Simplify the expression:
$$= \lim_{n \to \infty} \frac{5n^4}{n^4(1 + \frac{7}{n^4})} = \lim_{n \to \infty} \frac{5}{1 + \frac{7}{n^4}} = 5$$ Since the limit is a positive finite number (5), the Limit Comparison Test tells us that the original series $\sum_{n=1}^{\infty} \frac{5n^3}{n^4+7}$ behaves like the harmonic series.

Determine the convergence or divergence of the original series. Since the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges and the Limit Comparison Test shows that our series behaves like the harmonic series, the original series $\sum_{n=1}^{\infty} \frac{5n^3}{n^4+7}$ also diverges.
The series $\sum_{n=1}^{\infty} \frac{5n^3}{n^4+7}$ diverges.