Analysis of theorems on sequence limits:
Let $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$.

• Correct: By the fundamental limit theorem for sequences, if $(a_n)$ and $(b_n)$ converge, their sum $(a_n + b_n)$ converges and $\lim_{n \to \infty} (a_n + b_n) = A + B$.
• Incorrect: If $B \ne 0$, $(a_n / b_n)$ converges to $A / B$.
• Incorrect: If $a_n \to 0$, $(b_n / a_n)$ may diverge (e.g. division by zero limit).
• Incorrect: The difference $(a_n - b_n)$ is always convergent to $A - B$.

Mathematical evaluation of sequence limits:

• For sequence $(a_n) = n^2$:
  $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} n^2 = +\infty$$
  Since the limit is not a finite real number, $(a_n)$ is divergent.
• For sequence $(b_n) = 1 + \frac{1}{n^2}$:
  $$\lim_{n \to \infty} b_n = \lim_{n \to \infty} \left(1 + \frac{1}{n^2}\right) = 1 + 0 = 1$$
  Since the limit exists as a finite real number ($1$), $(b_n)$ is convergent.
• Incorrect: $(a_n)$ does not converge.
• Incorrect: $(b_n)$ does not diverge.
• Incorrect: The roles are inverted.
• Correct: $(a_n)$ is divergent and $(b_n)$ is convergent.

Step-by-step analysis of each option for $a_n = \frac{1}{\sqrt{n}} = n^{-1/2}$:

• Incorrect: By the Alternating Series (Leibniz) Test, since $a_n \gt 0$, $a_n$ is monotonically decreasing to $0$, $\sum (-1)^n a_n$ converges (conditionally).
• Incorrect: $\sum (a_n)^2 = \sum \frac{1}{n}$, which is the divergent harmonic series ($p = 1$).
• Correct: $\sum_{n=1}^\infty (a_n)^3 = \sum_{n=1}^\infty \left(\frac{1}{\sqrt{n}}\right)^3 = \sum_{n=1}^\infty \frac{1}{n^{3/2}}$. This is a standard $p$-series with $p = 3/2 \gt 1$, which is convergent.
• Incorrect: $\sum a_n = \sum \frac{1}{n^{1/2}}$ is a $p$-series with $p = 1/2 \le 1$, which is divergent.