The uploaded document contains a series of mathematical problems, mostly focusing on limits and sequences. Here's a breakdown of the content:

1. Limit Problems: You are asked to compute several limits involving different types of sequences. For example:

   • $\lim_{n \to \infty} \frac{\ln(n)}{n}$
   • $\lim_{n \to \infty} \frac{\ln(\ln(n))^3}{\ln(n)}$
   • $\lim_{n \to \infty} \frac{n^n}{e^{n^2}}$
   • etc.

2. Convergence of a Sequence: The second problem seems to involve the convergence of a sequence defined by nested radicals, with a condition that $p > 1$.

3. Permutation and Series: Problem 3 asks for a necessary and sufficient condition for a particular series to exist, involving a permutation of integers.

4. Sum and Limit: In problem 4, there is an expression involving a sum over $am$ (with some conditions), followed by a limit as $n \to \infty$.

5. Euler-Mascheroni Constant: The final problem is to prove that the sum of the harmonic series minus the logarithm converges to the Euler-Mascheroni constant $\gamma$ as $n \to \infty$.

Would you like to work through specific problems from this list? Let me know which ones you're most interested in!

Here are some follow-up questions:

1. Do you need assistance with calculating specific limits, like the first problem $\lim_{n \to \infty} \frac{\ln(n)}{n}$?
2. Would you like a more detailed explanation of how to approach problem 5 (convergence to $\gamma$)?
3. Would you like me to explain the approach to the nested radicals problem from problem 2?
4. Is problem 3 about permutations and series clear, or would you like more clarification on it?
5. Should I elaborate on the steps for evaluating limits with logarithmic functions?

Tip: When solving limits, especially for sequences involving logarithmic and exponential functions, L'Hopital's Rule and asymptotic approximations are often very useful techniques!