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Math Help - convergences

  1. #1
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    convergences

    -working in real numbers, true or false? proof or counterexample:
    (a) If a_n is decreasing sequence of positive numbers, and na_n \rightarrow 0 as n \rightarrow \infty, then \sum a_n converges.

    (b)If \sum a_n converges, then \sum \frac{a_n}{\sqrt{n}} converges.


    (c)If \sum a_n converges, then \sum \frac{|a_n|}{n} converges.


    (d)If \sum a_n converges, then \sum \frac{|a_n|}{n^{3/2}} converges.
    Last edited by mr fantastic; October 21st 2010 at 06:27 PM. Reason: Restored deleted question.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by DontKnoMaff View Post
    working in real numbers, true or false? proof or counterexample:
    (a) If a_n is decreasing sequence of positive numbers, and na_n \rightarrow 0 as n \rightarrow \infty, then \sum a_n converges.
    Consider:

    a_n=\frac{1}{n\ln(n)}

    \displaystyle \lim_{n\to \infty} na_n=\frac{1}{\ln(n)}=0

    but:  \sum_{n=2}^{\infty}a_n diverges (you might have an interesting time proving this )

    CB
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  3. #3
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    could compare it to  \sum_{n=1}^{\infty} \frac{1}{n} .
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  4. #4
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    Quote Originally Posted by Time View Post
    could compare it to  \sum_{n=1}^{\infty} \frac{1}{n} .
    NO since \frac{1}{n\ln(n)}<\frac{1}{n} unless you have something mor sophisticated in mind.

    CB
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