## anonymous 3 years ago i have the sequence a_n=( sin 1!/1*2)+( sin 2!/2*3)+( sin 3!/3*4) + ... + ( sin n!/n*(n+1)) i must use cauchy (a_(n+p) - a_n) to verify if it converges or not

1. anonymous

$a_n=\frac{ sin (1!)}{1*2}+\frac{ sin (2!)}{2*3}+....+\frac{ sin( n!)}{n*(n+1)}$ This?

2. anonymous

yes, this is it

3. anonymous

4. anonymous

you can ignore the sine part (i think) since it is bounded above by 1and below by -1

5. anonymous

$a_{n+p}-a_{n}=\frac{\sin((n+1)!)}{(n+1)(n+2)}+\frac{\sin((n+2)!)}{(n+2)(n+3)}+\cdots+\frac{\sin((n+p)!)}{(n+p)(n+p+1)}$

6. anonymous

$|a_{n+p}-a_{n}|\leq \frac{1}{(n+1)(n+2)}+\frac{1}{(n+2)(n+3)}+\cdots+\frac{1}{(n+p)(n+p+1)}$ which for n big enough is less than any epsilon

7. anonymous

got it?

8. anonymous

yup, thak you very much