https://file.aiquickdraw.com/mathbot/file/1730757233094qqym1csg.jpg

The problem given is:

\[
\sum_{n=0}^{\infty} \frac{2n + 5}{(n+2)^2 (n+3)^2}
\]

We need to determine whether this series converges or diverges and, if it converges, find its sum.

### Step 1: Analyze Convergence
To analyze the convergence of this series, we can consider applying the **comparison test** or **limit comparison test** with a simpler series that resembles the given one. Specifically, since the denominator grows as \(n^4\) (due to \((n+2)^2(n+3)^2\)), we might want to compare this with a known convergent series that has a similar structure.

### Step 2: Simplify the Growth of Terms
For large values of \(n\), the terms can be approximated as:

\[
\frac{2n + 5}{(n+2)^2 (n+3)^2} \approx \frac{2n}{n^4} = \frac{2}{n^3}
\]

So, we can compare this series to \(\sum \frac{1}{n^3}\), which is a convergent \(p\)-series with \(p = 3 > 1\). This suggests that our original series might also converge.

### Step 3: Applying the Limit Comparison Test
To confirm convergence, let's apply the **limit comparison test** with the series \(\sum \frac{1}{n^3}\):

\[
\lim_{n \to \infty} \frac{\frac{2n + 5}{(n+2)^2 (n+3)^2}}{\frac{1}{n^3}} = \lim_{n \to \infty} \frac{(2n + 5) n^3}{(n+2)^2 (n+3)^2}
\]

Expanding the denominator for large \(n\):

\[
(n+2)^2 \approx n^2, \quad (n+3)^2 \approx n^2 \Rightarrow (n+2)^2 (n+3)^2 \approx n^4
\]

Thus, the expression becomes:

\[
\lim_{n \to \infty} \frac{(2n + 5) n^3}{n^4} = \lim_{n \to \infty} \frac{2n + 5}{n} = \lim_{n \to \infty} 2 + \frac{5}{n} = 2
\]

Since this limit is a finite non-zero constant, the limit comparison test tells us that our series behaves similarly to \(\sum \frac{1}{n^3}\), which is convergent. Therefore, the original series **converges**.

### Step 4: Finding the Sum
The series converges, but finding an exact sum is more complex. This series does not easily simplify to a known form with a closed sum. Therefore, we conclude that the series **converges**, but without a specific closed-form sum available from standard series techniques.

### Final Answer
- **Convergence**: Yes, the series converges.
- **Sum**: Not easily obtainable in a closed form.

Would you like to see further details, or do you have any additional questions?

---

Here are some related questions to explore:

1. How does the \(p\)-series test determine convergence?
2. What other convergence tests could we apply to this series?
3. Why does the limit comparison test work in determining convergence?
4. How would the behavior of the series change if the exponent in the denominator were different?
5. Is there a way to approximate the sum of the series using numerical methods?

**Tip**: When unsure about convergence, try to approximate the behavior of the series for large \(n\) and compare it to known series types, such as \(p\)-series or geometric series.

Determine if the series \( \sum_{n=0}^{\infty} \frac{2n + 5}{(n+2)^2 (n+3)^2} \) converges or diverges and, if it converges, find the sum.

Learn how to determine the convergence of the series \( \sum_{n=0}^{\infty} \frac{2n + 5}{(n+2)^2 (n+3)^2} \) using the Limit Comparison Test. This solution covers convergence tests and the application of the P-Series Test. Suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation