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Math Help - Finding examples of sequences.

  1. #1
    Member ilikedmath's Avatar
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    Exclamation Finding examples of sequences.

    I am currently in an intro to real analysis class, and we are on limits now. The last time I had to deal with limits was in my last calculus class two years ago. So I am very rusty when it comes to limits. Any help, hints, suggestions, and/or corrections is greatly appreciated. Thank you for your time!

    (A) Give an example of a bounded sequence that diverges.

    * In class we already had the example of the sequence (-1)^n . Since it keeps bouncing back and forth.
    I was thinking maybe a sequence dealing with a trig function say sin x or cos x maybe?

    I'm stuck on B and C.

    (B) Give an example of a sequence of positive real numbers, { a_{n}}, where { a_{n}} converges and .


    (C) Give an example of a sequence of positive real numbers, { a_{n}}, such that but { a_{n}} diverges.
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  2. #2
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    b) a_n = \frac{1}{n}

    c) a_n = n
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  3. #3
    Member ilikedmath's Avatar
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    Quote Originally Posted by Plato View Post
    b) a_n = \frac{1}{n}
    Thanks, Plato!
    So I wrote out terms in this sequence to get {1, 1/2, 1/3, 1/4, ...}
    So if I did a_{n+1}/ a_{n}, I get the terms:
    {1/2, 2/3, 3/4, ...} which I can see eventually converges to one since the numerator is always one less than the denominator. However, how can I show algebraically that the limit = 1? And also how can I show algebraically or with theorems that the series {1/n} converges? It's harmonic, right?

    Quote Originally Posted by Plato View Post
    c) a_n = n
    Terms in this sequence: {1,2,3,...}
    So if I did a_{n+1}/ a_{n}, I get:
    {2,3/2,4/3,5/4,...} which I can 'see' diverges since the numerator is always one more than the denominator. However, how can I show algebraically that the limit is still = 1? And also how can I show algebraically or with theorems that the series {1/n} diverges?

    Thanks again!
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  4. #4
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    When I provided the examples, I assumed that you knew basic limits of sequences.
    Part (b) for this:
    a_n  = \frac{1}{n} \Rightarrow \quad \left( {a_n } \right) \to 0\,\& \,\left( {\frac{{a_{n + 1} }}{{a_n }}} \right) = \left( {\frac{n}{{n + 1}}} \right) \to 1.

    Part (c) is:
    b_n  = n \Rightarrow \quad \left( {b_n } \right) \to \infty \,\& \,\left( {\frac{{b_{n + 1} }}<br />
{{b_n }}} \right) = \left( {\frac{{n + 1}}<br />
{n}} \right) \to 1.
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  5. #5
    Member ilikedmath's Avatar
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    Quote Originally Posted by Plato View Post
    When I provided the examples, I assumed that you knew basic limits of sequences.
    Thank you very much for your patience and clarifications.
    I understand now. It took a while for me to refresh what I'd learned from calculus days.

    As for a bounded sequence that diverges, my example is {cos (n \pi)}. Since this sequence is bounded by [-1, 1] but the terms alternate between -1 and 1 therefore the sequence diverges.

    Is my explanation correct?
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