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I need to find all partial limits of$a_n=1+\frac{n+1}{n+3}\cos\frac{n\pi}{2} $, how I suppose to know what subsequences exactly to choose?

Thanks a lot!

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    What does partial limit mean ? Does it mean all cluster points ?2012-11-17
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    @Amr I mean limits of subsequences2012-11-17

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Hint: $\cos(\frac{n\pi}{2})=0$ for odd n, $cos(\frac{n\pi}{2})=-1$ for $n=2 (mod4) $, $cos(\frac{n\pi}{2})=1$ for $n=0 (mod 4)$

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    Why do you choose mode 4, not 5,6 or any else number why 4?Thanks.2012-11-17
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    OK. I will put this in my answer.2012-11-17
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    @ Amr Thanks, waiting for your explain2012-11-18