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I Divergence/Convergence for Telescoping series

  1. Mar 25, 2017 #1
    Can I use the divergence test on the partial sum of the telescoping series?
    Lim n>infinity an if not equal zero then it diverges

    The example below shows a telescoping series then I found the partial sum and took the limit of it. My question is shouldn't the solution be divergent? Since the result -1+cos 1 is not equal to 0? I'm confused.

    upload_2017-3-25_11-45-47.png
     
  2. jcsd
  3. Mar 25, 2017 #2
    that is right i guess
     
  4. Mar 25, 2017 #3
    it is ##a_n## must tend to zero, not the sum :)
     
  5. Mar 25, 2017 #4
    what's ##a_n##?
     
  6. Mar 25, 2017 #5
    a term of the series
     
  7. Mar 25, 2017 #6
    How can I determine convergence/divergence for telescopinc series then?
     
  8. Mar 25, 2017 #7
    you have already done this
     
  9. Mar 25, 2017 #8
    I haven't? the solution is above, however I don't quite understand. I want a general rule to detect convergence/divergence fore telescoping series.
     
  10. Mar 25, 2017 #9
    The series ##\sum (b_{n+1}-b_n)## is convergent iff the sequence ##b_n## is convergent
     
  11. Mar 25, 2017 #10
    but he didn't find the convergence of bn sepeartly in the solution above?
     
  12. Mar 25, 2017 #11
    He restored the proof of my proposition for this concrete example
     
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