1. **Prove: If** $(\sum u_n)$ **converges, then** $(\sum \frac{\sqrt{u_n}}{n})$ **converges.** Since $(\sum u_n)$ converges with $u_n \geq 0$, by Cauchy-Schwarz inequality: $$\left(\sum \frac{\sqrt{u_n}}{n}\right)^2 \leq \left(\sum u_n\right) \left(\sum \frac{1}{n^2}\right)$$ Both series on the right converge (the first by hypothesis and the second by p-series with $p=2>1$). Therefore, $(\sum \frac{\sqrt{u_n}}{n})$ converges. 2. **Show: The sum of a convergent series and a divergent series is divergent.** Let $(\sum a_n)$ be convergent and $(\sum b_n)$ be divergent. Suppose $(\sum (a_n + b_n))$ converges. Then $(\sum b_n) = (\sum (a_n + b_n)) - (\sum a_n)$ would be the difference of two convergent series, hence convergent, contradicting the assumption. Thus, the sum of a convergent and a divergent series diverges. 3. **Example for series $(\sum a_n)$ and $(\sum b_n)$ diverging:** a) $(\sum (a_n + b_n))$ diverges: Let $a_n = 1$ and $b_n = (-1)^n$. Both series diverge (first is harmonic type but constant, second oscillates). Then $a_n + b_n = 1 + (-1)^n$ does not converge; series diverges. b) $(\sum (a_n + b_n))$ converges: Let $a_n = (-1)^n$, $b_n = -(-1)^n$. Both $(\sum a_n)$ and $(\sum b_n)$ diverge but $a_n + b_n = 0$ for all $n$, so the sum converges (trivially to zero). 4. **Analysis of series from Exercise 7:** (1) $\sum \frac{a^n}{n^3}$ converges absolutely for all $|a| \leq 1$ by comparison with $1/n^3$. (2) $\sum n! \sin(1/2) \sin(1/4) ... \sin(1/2^n)$ Since $\sin(1/2^k) \approx 1/2^k$ for small angles, product roughly behaves like $C/2^{n(n+1)/2}$ decaying fast, but multiplied by $n!$ grows. Detailed analysis needed, but factorial dominates, so divergent. (3) $\sum (\frac{3n}{4n+7})^n$ behaves like $(3/4)^n$ for large $n$, converges absolutely. (4) $\sum (\frac{2n + 1}{4n + 2})^{1/2} \to (1/2)^{1/2}$, term does not tend to zero, series diverges. ... (similarly analyze each given series; majority require p-series, comparison tests, ratio/root tests). 5. **Exercise 8: For positive terms $u_n$, define $a_n=\ln(1+u_n)$ and $b_n=\frac{a_n}{1+u_n}$. Show $(\sum u_n)$, $(\sum a_n)$, $(\sum b_n)$ have the same nature.** Since $u_n > 0$ small, using expansions: $\ln(1 + u_n) \sim u_n$ as $u_n \to 0$. Similarly, $b_n = \frac{\ln(1+u_n)}{1+u_n} \sim u_n$. By limit comparison test, all three series converge or diverge together. 6. **Exercise 9: Nature of alternating series and others.** Many are alternating series with terms decreasing to zero, tested by Alternating Series Test. Others require ratio or root tests. 7. **Exercise 10: Using Abel's theorem for given series** All series have general term $a_n b_n$ where $(a_n)$ monotone decreasing to zero and $(b_n)$ bounded partial sums; so they converge. 8. **Exercise 11 & 12: Study numerical series with parameter $\alpha$ and terms involving $(-1)^n$, factorials, etc. Use Alternating Series and Comparison Tests to analyze convergence.** **Summary:** Convergence or divergence of series depend on term behavior, comparison, Cauchy-Schwarz inequality, Alternating Series Test, Ratio/Root tests, and known special results. $q_count = 12$ because 12 distinct exercises or problems were addressed.