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Math Help - Proving the convergence of a series?

  1. #1
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    Proving the convergence of a series?

    (a) Explain why, when proving the convergence of a series
    ∑ of a(n) between n=0 and ∞ it is permissible to ignore
    finitely many terms of the series.

    My Answer:
    Let s(n)=∑ of a(n) between k=0 and n be the nth partial
    sum. The series ∑ of a(n) between k=0 and ∞
    converges iff the sequence (s(n)) of partial sums converges.

    Now suppose that t(n)=∑ of a(n) between k=m and n
    is the partial sum formed by ignoring the first m entries
    and t(n)→l∈ℝ as n→∞.
    Then s(n)=a(0)+a(1)++a(m-1)+t(n)
    and s(n)→a(0)+a(1)++a(m-1)+l
    by continuity of the operation of adding the constant value
    a(0)+a(1)++a(m-1). So the
    original series also converges.



    (b) Use (a) together with a form of the comparison test to show
    that the series ∑ of 1/(2n-5)(2n-3)(2n-3) between n=0 and ∞
    converges.


    Note: I have been told my first answer is correct, and could some also plz tell me who to post using the normal mathematical symbols, such as the sum of and a small n. Thankyou
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  2. #2
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    Quote Originally Posted by kbartlett View Post
    (a) Explain why, when proving the convergence of a series
    ∑ of a(n) between n=0 and ∞ it is permissible to ignore
    finitely many terms of the series.

    My Answer:
    Let s(n)=∑ of a(n) between k=0 and n be the nth partial
    sum. The series ∑ of a(n) between k=0 and ∞
    converges iff the sequence (s(n)) of partial sums converges.

    Now suppose that t(n)=∑ of a(n) between k=m and n
    is the partial sum formed by ignoring the first m entries
    and t(n)→l∈ℝ as n→∞.
    Then s(n)=a(0)+a(1)+…+a(m-1)+t(n)
    and s(n)→a(0)+a(1)+…+a(m-1)+l
    by continuity of the operation of adding the constant value
    a(0)+a(1)+…+a(m-1). So the
    original series also converges.



    (b) Use (a) together with a form of the comparison test to show
    that the series ∑ of 1/ between n=0 and ∞
    converges.


    Note: I have been told my first answer is correct, and could some also plz tell me who to post using the normal mathematical symbols, such as the sum of and a small n. Thankyou

    \sum_{n=3}^{\infty}\frac{1}{(2n-5)(2n-3)(2n-3)} \le \sum_{n=3}^{\infty}\frac{1}{(2n)^3}
    this is a p-series with p=3. so it converges.

    as for the La Tex code try this page
    Help : Displaying a formula
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