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Math Help - Sequence and Series Problems

  1. #1
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    Sequence and Series Problems

    Hello everyone

    I'm having trouble with these three questions and would like some help

    the questions are :

    (a) Can you give an example of a sequence which is increasing,
    bounded from above and divergent? Justify your answer.


    (b) Can you give an example of a convergent sequence {an} such that
    sum ak , k=0 to infinty is divergent? Justify your answer.

    (c) Can you give an example of a divergent sequence {an} such that
    sum ak , k=0 to infinty is convergent? Justify your answer.

    for the first question i thought of (-1)^n but its bounded from above and below and can't prove that it is increasing

    for second question i think 1/n is the answer , as the sequence converges to 0 but the series diverges because it's a harmonic series

    for the third question i have no idea at all

    Thanks in advance !
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  2. #2
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    Re: Sequence and Series Problems

    b) The harmonic series is divergent while its terms converge to 0.

    c) No, in order for a series to be convergent, the terms HAVE to converge to 0.
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    Re: Sequence and Series Problems

    Quote Originally Posted by Lawati9 View Post
    the questions are :
    (a) Can you give an example of a sequence which is increasing,
    bounded from above and divergent? Justify your answer.

    (b) Can you give an example of a convergent sequence {an} such that
    sum ak , k=0 to infinty is divergent? Justify your answer.

    (c) Can you give an example of a divergent sequence {an} such that
    sum ak , k=0 to infinty is convergent? Justify your answer.

    for the first question i thought of (-1)^n but its bounded from above and below and can't prove that it is increasing

    for second question i think 1/n is the answer , as the sequence converges to 0 but the series diverges because it's a harmonic series

    for the third question i have no idea at all
    a) There is a theorem: A bounded monotone sequence converges.

    b) You are correct.

    c) There is a theorem: \sum\limits_{n = 1}^\infty  {{a_n}} converges only if (a_n)\to 0~.
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