Recitation 20

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 $\lim_{n\to\infty}n^{1/n}=1$.