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Math Help - Which tests to use for convergence?

  1. #1
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    Which tests to use for convergence?

    Hey all i was wondering what tests to use for these series, as some of them i have no idea how to tackle, hopefully you can help me out .



    Pointing me in the right direction will help me greatly for my exam tomorrow!
    Thanks people.

    I had no idea how to attempt any of them, and my lecture notes aint helping me too. For the sin ones, would i use a comparison with 1/n? because its sin (n) ?
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  2. #2
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    Quote Originally Posted by simpleas123 View Post
    Hey all i was wondering what tests to use for these series, as some of them i have no idea how to tackle, hopefully you can help me out .



    Pointing me in the right direction will help me greatly for my exam tomorrow!
    Thanks people.

    I had no idea how to attempt any of them, and my lecture notes aint helping me too. For the sin ones, would i use a comparison with 1/n? because its sin (n) ?

    1) \left|\frac{n-\sin^2n}{n^3}\right|\leq \frac{n+1}{n^3}\leq 2\frac{1}{n^2} ...comparison test for absolute convergence

    2) \frac{1}{2^n+1}\leq \left(\frac{1}{2}\right)^n ...comparison test

    3) \lim_{n\to\infty}\frac{n}{\ln n} \neq 0 ... in fact, \frac{n}{\ln n}\xrightarrow [n\to\infty]{}\infty

    4) \frac{n+1/n}{n!+n^3}\leq \frac{2n}{n!}=2\frac{1}{(n-1)!} , and then use the ratio test to show this last series converges.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    1) \left|\frac{n-\sin^2n}{n^3}\right|\leq \frac{n+1}{n^3}\leq 2\frac{1}{n^2} ...comparison test for absolute convergence

    2) \frac{1}{2^n+1}\leq \left(\frac{1}{2}\right)^n ...comparison test

    3) \lim_{n\to\infty}\frac{n}{\ln n} \neq 0 ... in fact, \frac{n}{\ln n}\xrightarrow [n\to\infty]{}\infty

    4) \frac{n+1/n}{n!+n^3}\leq \frac{2n}{n!}=2\frac{1}{(n-1)!} , and then use the ratio test to show this last series converges.

    Tonio
    Thank you very much! +rep'd
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