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## graydarl 2 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

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1. henpen

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

2. graydarl

yes, this is it

3. graydarl

@henpen help me please :D

4. satellite73

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

5. myko

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

6. myko

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

7. myko

got it?

8. graydarl

yup, thak you very much

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