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Short problem, but why is this the answer? Confused

  1. Mar 12, 2012 #1
    Short problem, but why is this the answer? Confused :(

    Suppose [itex]\sum_{n=1}^\infty a_n[/itex] converges. Determine the convergence of [itex]\sum_{n=1}^\infty a_n+4[/itex]

    The answer is "divergent," but I don't see why that's necessarily true. I would assume we wouldn't know whether a_n + 4 is convergent/divergent.
     
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  3. Mar 12, 2012 #2

    micromass

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    Re: Short problem, but why is this the answer? Confused :(

    If [itex]\sum_{n=1}^{+\infty} a_n[/itex] converges, then what is [itex]\lim_{n\rightarrow +\infty}{ a_n}[/itex]??
     
  4. Mar 12, 2012 #3
    Re: Short problem, but why is this the answer? Confused :(

    If the first series you have there converges, the second one also converges. Adding or subtracting a finite number from a convergent series cannot make it divergent.
     
  5. Mar 12, 2012 #4
    Re: Short problem, but why is this the answer? Confused :(

    @micromass, the lim would be a finite number?
     
  6. Mar 12, 2012 #5
    Re: Short problem, but why is this the answer? Confused :(

    Remember the test for divergence, if a series converges, the lim of the sequence must be zero at infinity.
     
  7. Mar 12, 2012 #6
    Re: Short problem, but why is this the answer? Confused :(

    Ok, that makes sense. Does this prove that my answer is correct, or does the book answer still hold?
     
  8. Mar 12, 2012 #7

    micromass

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    Re: Short problem, but why is this the answer? Confused :(

    Depends on whether you mean

    [tex]\sum (a_n + 4)[/tex]

    or

    [tex]\left(\sum a_n\right) +4[/tex]
     
  9. Mar 12, 2012 #8
    Re: Short problem, but why is this the answer? Confused :(

    The first one, which I assume makes the book answer correct.
     
  10. Mar 12, 2012 #9

    micromass

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    Re: Short problem, but why is this the answer? Confused :(

    OK. For the first one, what is the limit of the term sequence?? What is

    [tex]\lim_{n\rightarrow +\infty}{a_n+4}[/tex]

    ??
     
  11. Mar 12, 2012 #10
    Re: Short problem, but why is this the answer? Confused :(

    I'd think it would be 4?
     
  12. Mar 12, 2012 #11
    Re: Short problem, but why is this the answer? Confused :(

    I guess I was right lol
     
  13. Mar 12, 2012 #12

    micromass

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    Re: Short problem, but why is this the answer? Confused :(

    Yes!! So can the series possibly converge?? Remember that if it converged then the limit would be 0.
     
  14. Mar 12, 2012 #13
    Re: Short problem, but why is this the answer? Confused :(

    Ahh, ok, now I understand. Thank you for the help :D
     
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