GST, nth term, factorial

simplify  $\frac{n!}{\left(n+1\right)!}$  

•  $\frac{1}{1!}$  

•  $\frac{1}{n+1}$  

•  $n+1$  

•  $\frac{1}{\left(n+1\right)!}$  

simplify  $\frac{5!}{3!}$  

• 20

•  $\frac{5}{3}$  

• 9

• 120

simplify  $\frac{6!}{3!\ 2!}$  

•  $6\cdot5\cdot2=60$  

•  $5!=120$  

•  $1!=1$  

•  $6$  

Tags

CCSS.HSA.APR.D.6

CCSS.HSA.APR.D.7

simplify $\frac{2\left(3!\right)}{6}$  

•  $2$  

•  $5!=120$  

•  $1$  

•  $\frac{1}{6}$  

Tags

CCSS.7.NS.A.2

CCSS.7.NS.A.3

simplify  $\frac{\left(n+2\right)!}{\left(n-1\right)!}$  

•  $n^3+3n^2+2n$  

•  $3\cdot n!$  

•  $\left(n+2\right)\left(n+1\right)$  

•  $\left(n+2\right)!$  

 $\left(2n\right)!=$  

•  $2n\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)...$  

•  $2\cdot n\left(n-1\right)\left(n-2\right)\left(n-3\right)...$  

•  $2^n\cdot n!$  

•  $\left(n!\right)^2$  

 $\sum_{n=1}^{\infty}5\left(\frac{2}{3}\right)^{n-1}$  

• converges by GST because  $\left|r\right|=\frac{2}{3}$  
  

• diverges by GST because $\left|r\right|=\frac{2}{3}$  

• diverges by GST because $\left|r\right|=5$  

• converges by GST because  $\left|a_1\right|=\frac{10}{3}$  

 $\sum_{n=1}^{\infty}\frac{1}{4}\left(-\frac{3}{2}\right)^n$  

• diverges by GST because  $\left|r\right|=\frac{3}{2}$  

• converges by GST because  $\left|r\right|=\frac{3}{2}$  

• diverges by GST because  $a_1=-\frac{3}{8}$  

• converges by GST because  $r=\frac{1}{4}$  

 $\sum_{n=1}^{\infty}\left(\sqrt{2}\right)^{-n}$  

• converges by GST

• diverges by GST

• neither GST nor nth term test applies

• diverges by the nth term test

 $\sum_{n=1}^{\infty}\frac{n}{n+5}$  

•  $\lim_{n\rightarrow\infty}\ \frac{n}{n+5}=1\ \ \therefore$  series diverges by nth term 

•  $\lim_{n\rightarrow\infty}\ \frac{n}{n+5}=1\ \ \therefore$  the nth term test is inconclusive 

•  $\lim_{n\rightarrow\infty}\ \frac{n}{n+5}=1\ \ \therefore$  series converges by nth term 

•  $\left|r\right|=1$  series diverges by nth GST