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Infinite series

  1. Dec 13, 2011 #1
    infinite series!!

    1. The problem statement, all variables and given/known data
    Hi!!
    I've learned that the definition of a sequence of elements of complex numbers is as follows;
    a sequence is a function whose domain is a set of all positive integers with values in a set consisting of all complex numbers. (Denote the set of all complex numbers by C from now on)
    Now, let {a[itex]_{n}[/itex]} be a sequence of elements of C.
    Then, as you know, infinite series is defined as a sequence of partial sums b[itex]_{k}[/itex] = [itex]\sum[/itex][itex]^{k}_{n=1}[/itex]a[itex]_{n}[/itex].If the limit of the sequence exists, then it is said that the infinite series{b[itex]_{k}[/itex]} is convergent. In this case,
    a value of the limit of the sequence is called a sum of the series and is denoted by
    lim[itex]_{k\rightarrow}[/itex][itex]\infty[/itex]b[itex]_{k}[/itex].

    Now, here is my question.
    I've seen a notation like this; [itex]\sum[/itex][itex]_{n=p}[/itex][itex]^{\infty}[/itex]a[itex]_{n}[/itex] where p is any integer. If p is not 1, then I have no idea how to interpret this expression.....
    What is the exact definition of a kind of a sum above..?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 13, 2011 #2

    Dick

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    Re: infinite series!!

    It's [itex]a_p+a_{p+1}+a_{p+2}+...[/itex] It's the difference between the sum of the whole series and the (p-1)th partial sum.
     
  4. Dec 13, 2011 #3
    Re: infinite series!!

    Then what can you say about ,for example,[itex]\sum[/itex][itex]^{\infty}_{n=-3}[/itex]a[itex]_{n}[/itex]??
    Do you think that it is a difference between the sum of a sequence a[itex]_{1}[/itex],a[itex]_{2}[/itex],... and (a[itex]_{-3}[/itex]+...+a[itex]_{0}[/itex])?

    I want to know a kind of a definition such as a form of limit expression...as I showed in the case of [itex]\sum[/itex][itex]^{\infty}_{n=1}[/itex]a[itex]_{n}[/itex].
     
  5. Dec 13, 2011 #4

    Dick

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    Re: infinite series!!

    You can write a series starting at any value of n, like p in the example you give. If you take a_n=1/(2^n), then the sum from n=1 to infinity is 1. The sum from n=0 to infinity is 2. The sum from n=3 to infinity is 1/4. They only differ from the sum from n=1 to infinity by a finite number of terms, as you said. If a series converges then the series starting a different limits will also converge. Just to a slightly different limit. There's no law that says the first term of a series must be a_1.
     
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