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    The question is to determine the range of z for which the series (1 + sin n)^n *(z^n) is convergent.

    I perform the ratio test and calculate |an| ^(1/n) = (1 + sin n) |z|

    From this I deduce the limit superior of (1+ sin n) to be 2 and therefore the series should be absolutely convergent if |z|<0.5.

    However the answer given says the series is absolutely convergent if |z| < 1.

    Where is my mistake, or is the given answer wrong?
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    (Original post by takeonme79)
    The question is to determine the range of z for which the series (1 + sin n)^n *(z^n) is convergent.

    I perform the ratio test and calculate |an| ^(1/n) = (1 + sin n) |z|
    You mean the root test?

    From this I deduce the limit superior of (1+ sin n) to be 2 and therefore the series should be absolutely convergent if |z|<0.5.

    However the answer given says the series is absolutely convergent if |z| < 1.

    Where is my mistake, or is the given answer wrong?
    I can't see any problem with what you've written, but I'm not the world's greatest analyst.

    I couldn't see how to check it with Wolfram Alpha, but I'd guess that it's possible.

    (BTW, it'd be easier to read what you've written if you use latex)
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    (Original post by atsruser)
    You mean the root test?



    I can't see any problem with what you've written, but I'm not the world's greatest analyst.

    I couldn't see how to check it with Wolfram Alpha, but I'd guess that it's possible.

    (BTW, it'd be easier to read what you've written if you use latex)
    You're right, I did mean the root test. I have to learn Latex some day, never got round to doing it so far.

    I'm fairly sure I'm right about my answer as I've used a calculator to calculate the sum of the first 100 and 500 terms if z=0.9 and that seems to diverge.
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    (Original post by takeonme79)
    The question is to determine the range of z for which the series (1 + sin n)^n *(z^n) is convergent.

    I perform the ratio test and calculate |an| ^(1/n) = (1 + sin n) |z|

    From this I deduce the limit superior of (1+ sin n) to be 2 and therefore the series should be absolutely convergent if |z|<0.5.

    However the answer given says the series is absolutely convergent if |z| < 1.

    Where is my mistake, or is the given answer wrong?
    You have not provided a series. Why don't you write the question in an unambiguous and accurate way and then ask for help?


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    (Original post by LightBlueSoldier)
    You have not provided a series. Why don't you write the question in an unambiguous and accurate way and then ask for help?
    I didn't think that "the series (1 + sin n)^n *(z^n) was that ambiguous myself.

    (Original post by takeonme79)
    ..
    To make sure we're talking about the same series, I'm assuming the exact series is:

    \displaystyle \sum_{n=0}^\infty (1+ \sin n)^n z^n,

    in which case I agree that it doesn't converge for |z| > 1/2, since \limsup |1+\sin n| = 2
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    (Original post by DFranklin)
    I didn't think that "the series (1 + sin n)^n *(z^n) was that ambiguous myself.

    ]
    Of course, but 99% of the time when someone has an answer different to the given one it is because they have misread or misunderstood the question. I've been guilty of it enough times myself to know that.


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