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Math Help - limits of sequences

  1. #1
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    limits of sequences

    a, Let q be the element of real numbers with q which is not equal to 0.
    Show that if the absolute of q is less than 1, then
    the limit of an absolute of q^n=0 as lim approaches positive infinity.

    b, As the absolute of q which is greater than 1 then the
    limit of an absolute of q^n =positive infinity as the limit approaches positive infinity.

    c, What is the value of x if the sum of n=2 to positive infinity of (1+x)^-n=2
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  2. #2
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    Quote Originally Posted by savetra View Post
    a, Let q be the element of real numbers with q which is not equal to 0.
    Show that if the absolute of q is less than 1, then
    the limit of an absolute of q^n=0 as lim approaches positive infinity.
    Use the ratio test,
    lim |q|^{n+1}/|q|^n=|q|<1
    Thus,
    The infinite series,
    SUM |q|^n---->Converges.
    But then the sequence must converge to zero,
    That it,
    lim |q|^n---->0
    ----
    b)

    If |q|>1
    Then multiplication by |q|>0 yields,
    |q|^2>|q|
    Again,
    |q|^3>|q|^2
    Again and again...
    Thus the sequence is not bounded.
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