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Math Help - quick question about sequences

  1. #1
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    quick question about sequences

    if (an) is a sequence and (bn) is a sequence, then what is the sequence max(an, bn). Is it (an) if (an) converges to a higher limit and (bn) if (bn) converges to a higher limit?

    also, can someone just tell me if the following statements are true or false? i'll prove them or come up with a counterexample, i just want to know which are true.

    1.) if (an) and (bn) are divergent, then also (an + bn) and (an - bn) are divergent. I believe this is false.
    2.) (an)^2 converges if and only if (abs(an)) converges. true?
    3.) if (an) and (bn) are convergent, then also max(an, bn) is convergent. true?
    4.) if (an+1 - an) converges to zero, then (an) converges. *the n+1 is all a subscript*
    5.) if an > 0 and ((an+1)/(an)) < 1 for all n, then (an) converges. *again the n+1 is all a subscript*
    I just want to know if they're true or false.
    Last edited by CindyMichelle; October 4th 2007 at 02:05 PM.
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  2. #2
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    Quote Originally Posted by CindyMichelle View Post
    if (an) is a sequence and (bn) is a sequence, then what is the sequence max(an, bn). Is it (an) if (an) converges to a higher limit and (bn) if (bn) converges to a higher limit?
    \max \{ a_n , b_n\} = \frac{1}{2}(|a_n+b_n| - |a_n-b_n|)
    Or something like that (I cannot think exactly what it is now).
    1.) if (an) and (bn) are divergent, then also (an + bn) and (an - bn) are divergent. I believe this is false.
    How about a_n = (-1)^n and b_n = (-1)^{n+1}.
    2.) (an)^2 converges if and only if (abs(an)) converges. true?
    Since this converges it must mean that \sqrt{a_n^2} = |a_n| coverges.
    3.) if (an) and (bn) are convergent, then also max(an, bn) is convergent. true?
    Look at first post.
    4.) if (an+1 - an) converges to zero, then (an) converges. *the n+1 is all a subscript*
    No consider the sequence of partial harmonic sums.
    5.) if an > 0 and ((an+1)/(an)) < 1 for all n, then (an) converges. *again the n+1 is all a subscript*
    I just want to know if they're true or false.
    Yes. And furthermore it converges to zero. This is the ratio test.
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  3. #3
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    thank you so much for answering.

    what's the sequence of partial harmonic sums? I'm not familiar with it. what's an example of one?
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    Quote Originally Posted by CindyMichelle View Post
    thank you so much for answering.

    what's the sequence of partial harmonic sums? I'm not familiar with it. what's an example of one?
    the Harmonic series \sum_{n = 1}^{\infty} \frac 1n = 1 + \frac 12 + \frac 13 + \frac 14 + ... is a VERY important series that you must know. it diverges, yet \sum_{n = 1}^{\infty} \left( \frac 1{n+1} - \frac 1n \right) converges to -1 (can you show that?)
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    It has to converge to zero. Does this still work?
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    h_n  = \sum\limits_{k = 1}^n {\frac{1}{k}} \quad  \Rightarrow \quad \left( {h_n } \right) \to \infty

    \left( {h_{n + 1}  - h_n } \right) = \frac{1}{{n + 1}}\quad  \Rightarrow \quad \left( {h_{n + 1}  - h_n } \right) \to 0<br />
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    h_n  = \sum\limits_{k = 1}^n {\frac{1}{k}} \quad  \Rightarrow \quad \left( {h_n } \right) \to \infty

    \left( {h_{n + 1}  - h_n } \right) = \frac{1}{{n + 1}}\quad  \Rightarrow \quad \left( {h_{n + 1}  - h_n } \right) \to 0<br />
    yes, i should have said the sum equals -1. so that means it converges. and if it converges, then the sequence describing the nth term goes to zero
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