5.4 Comparison tests  (Page 4/7)

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{2\left(n+1\right)}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{2n-1}$

${\displaystyle \sum }_{n=2}^{\infty }\frac{1}{{\left(n\text{ln}n\right)}^2}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{n\text{!}}{\left(n+2\right)\text{!}}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n\text{!}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{\text{sin}\left(1\text{/}n\right)}{n}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{{\text{sin}}^{2}n}{n^2}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{\text{sin}\left(1\text{/}n\right)}{\sqrt{n}}}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{n^{1.2}-1}{n^{2.3}+1}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{\sqrt{n+1}-\sqrt{n}}{n}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{\sqrt[4]{n}}{\sqrt[3]{n^4+n^2}}$

${\displaystyle \sum }_{n=1}^{\infty }{\left(\frac{\text{ln}n}{n}\right)}^2$

${\displaystyle \sum }_{n=1}^{\infty }{\left(\frac{\text{ln}n}{n^{0.6}}\right)}^2$

${\displaystyle \sum }_{n=1}^{\infty }\frac{\text{ln}\left(1+\frac{1}{n}\right)}{n}$

${\displaystyle \sum }_{n=1}^{\infty }\text{ln}\left(1+\frac{1}{n^2}\right)$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{4^n-3^n}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^2-n\text{sin}n}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{e^{\left(1.1\right)n}-3^n}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{e^{\left(1.01\right)n}-3^n}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^{1+1\text{/}n}}$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{2^{1+1\text{/}n}{n}^{1+1\text{/}n}}$

${\displaystyle \sum }_{n=1}^{\infty }\left(\frac{1}{n}-\text{sin}\left(\frac{1}{n}\right)\right)$

${\displaystyle \sum }_{n=1}^{\infty }\left(1-\text{cos}\left(\frac{1}{n}\right)\right)$

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n}\left({\text{tan}}^{\mathrm{-1}}n-\frac{\pi }{2}\right)$

${\displaystyle \sum }_{n=1}^{\infty }{\left(1-\frac{1}{n}\right)}^{n.n}$ ( Hint: ${\left(1-\frac{1}{n}\right)}^n\to 1\text{/}e.)$

${\displaystyle \sum }_{n=1}^{\infty }\left(1-e^{\mathrm{-1}\text{/}n}\right)$ ( Hint: $1\text{/}e\approx {\left(1-1\text{/}n\right)}^n,$ so $1-e^{\mathrm{-1}\text{/}n}\approx 1\text{/}n.)$

Does ${\displaystyle \sum }_{n=2}^{\infty }\frac{1}{{\left(\text{ln}n\right)}^p}$ converge if $p$ is large enough? If so, for which $p\text{?}$

Does ${\displaystyle \sum }_{n=1}^{\infty }{\left(\frac{\left(\text{ln}n\right)}{n}\right)}^p$ converge if $p$ is large enough? If so, for which $p\text{?}$

For which $p$ does the series ${\displaystyle \sum }_{n=1}^{\infty }{2}^{pn}\text{/}{3}^n$ converge?

For which $p>0$ does the series ${\displaystyle \sum }_{n=1}^{\infty }\frac{n^p}{2^n}$ converge?

For which $r>0$ does the series ${\displaystyle \sum }_{n=1}^{\infty }\frac{r^{n^2}}{2^n}$ converge?

For which $r>0$ does the series ${\displaystyle \sum }_{n=1}^{\infty }\frac{2^n}{r^{n^2}}$ converge?

Find all values of $p$ and $q$ such that ${\displaystyle \sum }_{n=1}^{\infty }\frac{n^p}{{\left(n\text{!}\right)}^q}$ converges.

Does ${\displaystyle \sum }_{n=1}^{\infty }\frac{{\text{sin}}^2\left(nr\text{/}2\right)}{n}$ converge or diverge? Explain.

Explain why, for each $n,$ at least one of $\{\left|\text{sin}n\right|,\left|\text{sin}\left(n+1\right)\right|\text{,...},\left|\text{sin}n+6\right|\}$ is larger than $1\text{/}2.$ Use this relation to test convergence of ${\displaystyle \sum }_{n=1}^{\infty }\frac{\left|\text{sin}n\right|}{\sqrt{n}}.$

Suppose that $a_n\ge 0$ and $b_n\ge 0$ and that ${\displaystyle \sum _{n=1}^{\infty }{a}^2{}_n}$ and ${\displaystyle \sum _{n=1}^{\infty }{b}^2{}_n}$ converge. Prove that ${\displaystyle \sum _{n=1}^{\infty }{a}_n{b}_n}$ converges and ${\displaystyle \sum _{n=1}^{\infty }{a}_n}{b}_n\le \frac{1}{2}\left({\displaystyle \sum _{n=1}^{\infty }{a}_n^2}+{\displaystyle \sum _{n=1}^{\infty }{b}_n^2}\right).$

Does ${\displaystyle \sum _{n=1}^{\infty }{2}^{\text{−}\text{ln}\text{ln}n}}$ converge? ( Hint: Write $2^{\text{ln}\text{ln}n}$ as a power of $\text{ln}n.)$

Does ${\displaystyle \sum _{n=1}^{\infty }{\left(\text{ln}n\right)}^{\text{−}\text{ln}n}}$ converge? ( Hint: Use $t=e^{\text{ln}\left(t\right)}$ to compare to a $p-\text{series}\text{.})$

Does ${\displaystyle \sum _{n=2}^{\infty }{\left(\text{ln}n\right)}^{\text{−}\text{ln}\text{ln}n}}$ converge? ( Hint: Compare $a_n$ to $1\text{/}n.)$

Show that if $a_n\ge 0$ and ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges, then ${\displaystyle \sum _{n=1}^{\infty }{a}^2{}_n}$ converges. If ${\displaystyle \sum _{n=1}^{\infty }{a}^2{}_n}$ converges, does ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ necessarily converge?