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graydarl

  • 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

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

  2. graydarl
    • 3 years ago
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    yes, this is it

  3. graydarl
    • 3 years ago
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    @henpen help me please :D

  4. anonymous
    • 3 years ago
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    you can ignore the sine part (i think) since it is bounded above by 1and below by -1

  5. myko
    • 3 years ago
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    \[a_{n+p}-a_{n}=\frac{\sin((n+1)!)}{(n+1)(n+2)}+\frac{\sin((n+2)!)}{(n+2)(n+3)}+\cdots+\frac{\sin((n+p)!)}{(n+p)(n+p+1)}\]

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

  7. myko
    • 3 years ago
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    got it?

  8. graydarl
    • 3 years ago
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    yup, thak you very much

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