Question

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

Answer 1

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

Answer 2

yes, this is it

Answer 3

@henpen help me please :D

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

got it?

Answer 8

yup, thak you very much