5.6 Ratio and root tests  (Page 5/8)

$a_n=n^{10}\text{/}{2}^n$

${\displaystyle \sum _{n=1}^{\infty }\frac{{(n\text{!})}^3}{(3n\text{!})}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{2^{3n}{(n\text{!})}^3}{(3n\text{!})}}$

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

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

${\displaystyle \sum _{n=1}^{\infty }\frac{n\text{!}}{{(n\text{/}e)}^n}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{(2n)\text{!}}{{(n\text{/}e)}^{2n}}}$

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

$a_k={\left(\frac{k-1}{2k+3}\right)}^k$

$a_k={\left(\frac{2k^2-1}{k^2+3}\right)}^k$

$a_n=\frac{{\left(\text{ln}n\right)}^{2n}}{n^n}$

$a_n=n\text{/}{2}^n$

$a_n=n\text{/}{e}^n$

$a_k=\frac{k^e}{e^k}$

$a_k=\frac{{\pi }^k}{k^{\pi }}$

$a_n={\left(\frac{1}{e}+\frac{1}{n}\right)}^n$

$a_k=\frac{1}{{\left(1+\text{ln}k\right)}^k}$

$a_n=\frac{{\left(\text{ln}\left(1+\text{ln}n\right)\right)}^n}{{\left(\text{ln}n\right)}^n}$

$a_k=\frac{k\text{!}}{1·3·5\text{⋯}(2k-1)}$

$a_k=\frac{2·4·6\text{⋯}2k}{(2k)\text{!}}$

$a_k=\frac{1·4·7\text{⋯}(3k-2)}{3^{k}k\text{!}}$

$a_n={\left(1-\frac{1}{n}\right)}^{n^2}$

$a_k={\left(\frac{1}{k+1}+\frac{1}{k+2}+\text{⋯}+\frac{1}{2k}\right)}^k$ ( Hint: Compare $a_k^{1\text{/}k}$ to ${\displaystyle {\int }_k^{2k}\frac{dt}{t}}.)$

$a_k={\left(\frac{1}{k+1}+\frac{1}{k+2}+\text{⋯}+\frac{1}{3k}\right)}^k$

$a_n={(n^{1\text{/}n}-1)}^n$

${\displaystyle \sum _{n=1}^{\infty }\frac{3^{n^2}}{2^{n^3}}}$

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

$a_n=1\text{/}{x}_n^n$ where $x_{n+1}=\frac{1}{2}{x}_n+\frac{1}{x_n},$ $x_1=1$ ( Hint: Find limit of $\left\{x_n\right\}.)$

${\displaystyle \sum _{n=1}^{\infty }\frac{\left(n+1\right)}{n^3+n^2+n+1}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(\mathrm{-1}\right)}^{n+1}\left(n+1\right)}{n^3+3n^2+3n+1}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(n+1\right)}^2}{n^3+{\left(1.1\right)}^n}}$

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

$a_n={\left(1+\frac{1}{n^2}\right)}^n$ ( Hint: ${\left(1+\frac{1}{n^2}\right)}^{n^2}\approx e.)$

$a_k=1\text{/}{2}^{{\text{sin}}^{2}k}$

$a_k=2^{\text{−}\text{sin}(1\text{/}k)}$

$a_n=1\text{/}\left(\begin{array}{c}n+2\\ n\end{array}\right)$ where $\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n\text{!}}{k\text{!}\left(n-k\right)\text{!}}$

$a_k=1\text{/}\left(\begin{array}{l}2k\\ k\end{array}\right)$

$a_k=2^k\text{/}\left(\begin{array}{l}3k\\ k\end{array}\right)$

$a_k={\left(\frac{k}{k+\text{ln}k}\right)}^k$ ( Hint: $a_k={\left(1+\frac{\text{ln}k}{k}\right)}^{\text{−}(k\text{/}\text{ln}k)\text{ln}k}\approx e^{\text{−}\text{ln}k}.)$

$a_k={\left(\frac{k}{k+\text{ln}k}\right)}^{2k}$ ( Hint: $a_k={\left(1+\frac{\text{ln}k}{k}\right)}^{\text{−}(k\text{/}\text{ln}k)\text{ln}{k}^2}.)$

${\displaystyle \sum _{k=1}^{\infty }\frac{k}{2^k}}$ ( Hint: Take $a_k=k$ and $b_k=2^{1-k}.)$

${\displaystyle \sum _{k=1}^{\infty }\frac{k}{c^k}},$ where $c>1$ ( Hint: Take $a_k=k$ and $b_k=c^{1-k}\text{/}(c-1).)$

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

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

${\displaystyle \sum _{k=1}^{\infty }\frac{x^k}{k^2}}$

${\displaystyle \sum _{k=1}^{\infty }\frac{x^{2k}}{k^2}}$

${\displaystyle \sum _{k=1}^{\infty }\frac{x^{2k}}{3^k}}$

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

Does there exist a number $p$ such that ${\displaystyle \sum _{n=1}^{\infty }\frac{2^n}{n^p}}$ converges?

Let $0<r<1.$ For which real numbers $p$ does ${\displaystyle \sum _{n=1}^{\infty }{n}^p}{r}^n$ converge?

Suppose that $\underset{n\to \infty }{\text{lim}}\left|\frac{a_{n+1}}{a_n}\right|=p.$ For which values of $p$ must ${\displaystyle \sum _{n=1}^{\infty }{2}^n}{a}_n$ converge?

Suppose that $\underset{n\to \infty }{\text{lim}}\left|\frac{a_{n+1}}{a_n}\right|=p.$ For which values of $r>0$ is ${\displaystyle \sum _{n=1}^{\infty }{r}^n}{a}_n$ guaranteed to converge?

Suppose that $\left|\frac{a_{n+1}}{a_n}\right|\le {(n+1)}^p$ for all $n=1,2\text{,…}$ where $p$ is a fixed real number. For which values of $p$ is ${\displaystyle \sum _{n=1}^{\infty }n}\text{!}{a}_n$ guaranteed to converge?