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Math Help - Cauchy sequences and series

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    Cauchy sequences and series

    What is the relationship between the Cauchy Criterion for sequences and Cauchy Criterion for series?
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    Quote Originally Posted by izzydoesit View Post
    What is the relationship between the Cauchy Criterion for sequences and Cauchy Criterion for series?
    The Cauchy Criterion for a sequence states that for EVERY epsilon > 0, there exists an N a natural number such than for n,m > N

    |a_n-a_m|< \epsilon

    The Key thing to remember is that a series is a sequence of partial sums


    The Cauchy Crterion for a series states that for EVERY epsilon > 0, there exists an N a natural number such than for n,m > N

    | \sum_{i=n}^{m}a_i| < \epsilon

    but what is this exactly? Well

    S_n=a_0+a_1+...a_n
    S_m=a_0+a_1+...a_n+a_{n+1}+...a_m

    These are both members of the sequence of partial sums

    |S_m-S_n|=|a_{n+1}+...a_m|=|\sum_{i=n}^{m}a_i|

    Now since we are talking about a sequence(of partial sums) we can use the Cauchy Criterion for sequences.

    I hope this clears it up.

    Good luck.
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