5.6 Ratio and root tests  (Page 6/8)

For which values of $r>0,$ if any, does ${\displaystyle \sum _{n=1}^{\infty }{r}^{\sqrt{n}}}$ converge? ( Hint: ${\displaystyle \sum _{n=1}^{\infty }{a}_n}={\displaystyle \sum _{k=1}^{\infty }{\displaystyle \sum _{n=k^2}^{{(k+1)}^2-1}{a}_n}}.)$

Suppose that $\left|\frac{a_{n+2}}{a_n}\right|\le r<1$ for all $n.$ Can you conclude that ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges?

Let $a_n=2^{\text{−}[n\text{/}2]}$ where $[x]$ is the greatest integer less than or equal to $x.$ Determine whether ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges and justify your answer.

Let $a_n=\frac{1}{4}\frac{3}{6}\frac{5}{8}\text{⋯}\frac{2n-1}{2n+2}=\frac{1·3·5\cdots (2n-1)}{2^n(n+1)\text{!}}.$ Explain why the ratio test cannot determine convergence of ${\displaystyle \sum _{n=1}^{\infty }{a}_n}.$ Use the fact that $1-1\text{/}(4k)$ is increasing $k$ to estimate $\underset{n\to \infty }{\text{lim}}\frac{a_{2n}}{a_n}.$

Let $a_n=\frac{1}{1+x}\frac{2}{2+x}\text{⋯}\frac{n}{n+x}\frac{1}{n}=\frac{(n-1)\text{!}}{(1+x)(2+x)\text{⋯}(n+x)}.$ Show that $a_{2n}\text{/}{a}_n\le e^{\text{−}x\text{/}2}\text{/}2.$ For which $x>0$ does the generalized ratio test imply convergence of ${\displaystyle \sum _{n=1}^{\infty }{a}_n}\text{?}$ ( Hint: Write $2a_{2n}\text{/}{a}_n$ as a product of $n$ factors each smaller than $1\text{/}\left(1+x\text{/}\left(2n\right)\right).)$

Let $a_n=\frac{n^{\text{ln}n}}{{\left(\text{ln}n\right)}^n}.$ Show that $\frac{a_{2n}}{a_n}\to 0$ as $n\to \infty .$

If $\underset{n\to \infty }{\text{lim}}{a}_n=0,$ then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges.

If $\underset{n\to \infty }{\text{lim}}{a}_n\ne 0,$ then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ diverges.

If ${\displaystyle \sum _{n=1}^{\infty }\left|a_n\right|}$ converges, then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges.

If ${\displaystyle \sum _{n=1}^{\infty }{2}^n{a}_n}$ converges, then ${\displaystyle \sum _{n=1}^{\infty }{\left(\mathrm{-2}\right)}^n{a}_n}$ converges.

$a_n=\frac{3+n^2}{1-n}$

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

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

$a_n=\frac{2^{n+1}}{5^n}$

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

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2+5n+4}}$

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

${\displaystyle \sum _{n=1}^{\infty }\frac{2^n}{n^4}}$

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

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

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

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

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

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

${\displaystyle \sum _{n=1}^{\infty }\text{cos}\left(\pi n\right)e^{\text{−}n}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{2^{n+4}}{7^n}}$

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

A legend from India tells that a mathematician invented chess for a king. The king enjoyed the game so much he allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, and so on. Find an exact expression for the total payment (in grains of rice) requested by the mathematician. Assuming there are $\mathrm{30,000}$ grains of rice in $1$ pound, and $2000$ pounds in $1$ ton, how many tons of rice did the mathematician attempt to receive?

Find $\underset{n\to \infty }{\text{lim}}{x}_n$ if $b>1,$ $b<1,$ and $b=1.$

Find an expression for $S_n={\displaystyle \sum _{i=0}^n{x}_i}$ in terms of $b$ and $x_0.$ What does it physically represent?

If $b=\frac{3}{4}$ and $x_0=100,$ find $S_{10}$ and $\underset{n\to \infty }{\text{lim}}{S}_n$

For what values of $b$ will the series converge and diverge? What does the series converge to?