**Example 1:** Suppose \sum a_n and \sum b_n are series with positive terms and **(a)** If a_n > b_n for all n, what can you say about a_n? Why? **(b)** If a_n < b_n for all n, what can you say about a_n? Why?

**Answer: (a)** If \sum b_n diverges, so does \sum a_n. **(b)** If \sum b_n converges, so does \sum a_n.

**Problem 2:** Determine whether the series is convergent or divergent? **(a)** \sum_{n=1}^\infty \frac{n+1}{n\sqrt{n}}; **(b)** \sum_{n=1}^\infty\frac{9^n}{3+10^n}; **(c)** \sum_{k=1}^\infty\frac{k\sin^2k}{1+k^3}; **(d)** \sum_{k=1}^\infty\frac{(2k-1)(k^2-1)}{(k+1)(k^2+4)^2}; **(e)** \sum_{n=1}^\infty\frac{4^{n+1}}{3^n-2}; **(f)** \sum_{n=1}^\infty\frac{1}{\sqrt{n^2+1}}; **(g)** \sum_{n=1}^\infty\frac{1}{n!}; **(h)** \sum_{n=1}^\infty\sin(1/n); **(i)** \sum_{n=1}^\infty\frac{1}{n^{1+1/n}}.

**Hint: (a)** Compare \frac{n+1}{n\sqrt{n}} with 1/\sqrt{n}; **(b)** Compare \frac{9^n}{3+10^n} with (9/10)^n; **(c)** Use \frac{k\sin^2k}{1+k^3} < 1/k^2; **(d)** Compare \frac{(2k-1)(k^2-1)}{(k+1)(k^2+4)^2} with 1/k^3; **(e)** Compare \frac{4^{n+1}}{3^n-2} with (4/3)^n; **(f)** Compare \frac{1}{\sqrt{n^2+1}} with 1/n; **(g)** Use 1/n! < 1/2^n for all n>1; **(h)** Compare \sin(1/n) with 1/n; **(i)** Compare 1/n^{1+1/n} with 1/n and use \lim_{n\to\infty}n^{1/n}=1.