The image contains three summation problems. Let's analyze and solve them one by one:

Problem (a):

$\sum_{n=1}^\infty \frac{2n + 1}{3^n + 1}$

This is an infinite series. To determine whether it converges or diverges:

1. Check the general term $\frac{2n + 1}{3^n + 1}$ as $n \to \infty$.

   • Since the denominator $3^n + 1$ grows much faster than the numerator $2n + 1$, the terms tend to $0$, which satisfies a necessary condition for convergence.

2. Perform a formal test:

   • Comparison Test: Compare with a simpler term such as $\frac{2n}{3^n}$, which can be rewritten as $\frac{2}{3^n/n}$. Since $3^n/n$ grows exponentially, $\frac{2}{3^n/n}$ converges, implying that the original series also converges.

Thus, the series converges. However, finding its exact sum would require additional tools or approximations.

Problem (b):

$\sum_{n=1}^\infty \left(\frac{4n + 1}{n^2 - 1}\right)^n$

1. Consider the general term $\left(\frac{4n + 1}{n^2 - 1}\right)^n$ as $n \to \infty$:

   • The ratio $\frac{4n + 1}{n^2 - 1}$ simplifies for large $n$ to approximately $\frac{4}{n}$.
   • Therefore, the term $\left(\frac{4n + 1}{n^2 - 1}\right)^n \sim \left(\frac{4}{n}\right)^n$.
   • For large $n$, $\left(\frac{4}{n}\right)^n$ approaches $0$ very rapidly.

2. Convergence Test:

   • Since the terms decay faster than any geometric progression, the series converges.

Problem (c):

$\sum_{n=1}^\infty n \cdot 3^{n^2}$

1. Analyze the general term $n \cdot 3^{n^2}$:

   • The term $3^{n^2}$ grows much faster than $n$ as $n \to \infty$.

2. Apply a test:

   • Using the comparison test, compare $n \cdot 3^{n^2}$ with $3^{n^2}$. Since both terms diverge to infinity, the series diverges.

Would you like step-by-step derivations for any of these solutions or additional insights? Let me know!

Here are five related questions to expand your understanding:

1. How does the ratio test help in determining the convergence of series?
2. What is the difference between absolute and conditional convergence?
3. Can you write a general test for comparing series involving factorials or exponential terms?
4. Why does exponential growth dominate polynomial growth in convergence analysis?
5. How would the results change if these summations started from $n = 0$ instead of $n = 1$?

Tip: Always simplify the general term first to identify dominant growth rates before applying convergence tests.