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Math Help - Real Analysis Help. Series Convergence

  1. #1
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    Real Analysis Help. Series Convergence

    i would love if someone could help me with 2 problems im having trouble with.


    1) Let a_1, a_2, a_3, ... be a decreasing sequence of positive numbers. Show that a_1+a_2+a_3+... converges if and only if a_1+2a_2+4a_4+8a_8+... converges. i saw a similar one on here a couple days ago but this is slightly different

    2) Show that a power series \sum\limits_{n = 1}^\infty {c_nx^n }\ has the same radius of convergence as \sum\limits_{n = 1}^\infty {c_{n+m}x^n }\, for any positive integer m

    thanks
    Last edited by megamet2000; November 20th 2008 at 04:36 PM. Reason: format
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  2. #2
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    Here is the problem: a_n > a_{n + 1} > 0\,,\,\sum\limits_{n = 1}^\infty {a_n } \,\& \,\sum\limits_{n = 0}^\infty {2^n a_{2^n } } .

    Using that a_n is decreasing, then following shows it in one direction.
    \begin{gathered} a_1 + \underbrace {a_2 + a_3 }_{} + \underbrace {a_4 + a_5 + a_6 + a_7 }_{} + \underbrace {a_8 + a_9 + a_{10} + a_{11} + a_{12} + a_{13} + a_{14} + a_{15} }_{} + \cdots \hfill \\<br />
\leqslant a_1 + 2a_2 + 4a_4 + 8a_8 \cdots \hfill \\ <br />
\end{gathered}

    Now note that:
    \begin{gathered} a_2 + \underbrace {a_3 + a_4 }_{} + \underbrace {a_5 + a_6 + a_7 + a_8 }_{} + \underbrace {a_9 + a_{10} + a_{11} + a_{12} + a_{13} + a_{14} + a_{15} + a_{16} }_{} + \cdots \hfill \\ \geqslant a_2 + 2a_4 + 4a_8 + 8a_{16}\cdots \hfill \\ \end{gathered}
    Also note that if \sum\limits_{n = 1}^\infty {a_n } converges that <br />
\sum\limits_{n = 2}^\infty {2a_n } converges.

    Can you finish?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Here is the problem: a_n > a_{n + 1} > 0\,,\,\sum\limits_{n = 1}^\infty {a_n } \,\& \,\sum\limits_{n = 0}^\infty {2^n a_{2^n } } .

    Using that a_n is decreasing, then following shows it in one direction.
    \begin{gathered} a_1 + \underbrace {a_2 + a_3 }_{} + \underbrace {a_4 + a_5 + a_6 + a_7 }_{} + \underbrace {a_8 + a_9 + a_{10} + a_{11} + a_{12} + a_{13} + a_{14} + a_{15} }_{} + \cdots \hfill \\<br />
\leqslant a_1 + 2a_2 + 4a_4 + 8a_8 \cdots \hfill \\ <br />
\end{gathered}

    Now note that:
    \begin{gathered} a_2 + \underbrace {a_3 + a_4 }_{} + \underbrace {a_5 + a_6 + a_7 + a_8 }_{} + \underbrace {a_9 + a_{10} + a_{11} + a_{12} + a_{13} + a_{14} + a_{15} + a_{16} }_{} + \cdots \hfill \\ \geqslant a_2 + 2a_4 + 4a_8 + 8a_{16}\cdots \hfill \\ \end{gathered}
    Also note that if \sum\limits_{n = 1}^\infty {a_n } converges that <br />
\sum\limits_{n = 2}^\infty {2a_n } converges.

    Can you finish?
    thanks for the help i appreciate it. proofs have never been my strong point would u mind helping me finish this?

    also can anyone help me with the second question?
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by megamet2000 View Post

    2) Show that a power series \sum\limits_{n = 1}^\infty {c_nx^n }\ has the same radius of convergence as \sum\limits_{n = 1}^\infty {c_{n+m}x^n }\, for any positive integer m

    thanks
    http://www.mathhelpforum.com/math-he...er-series.html
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  5. #5
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    awesome thank you very much
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  6. #6
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    does anyone have any tips on how to finish number 1?
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by megamet2000 View Post
    does anyone have any tips on how to finish number 1?
    What are you having problems with? Plato practicially gave you the solution.
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  8. #8
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    i dont understand the purpose of the braces. i get how a_n<2^na_{2^n} but after he says "now note that" i dont get how that sequence is now greater than the last sequence
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  9. #9
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    The first inequality tell us that \sum\limits_{k = 1}^\infty {a_k } \leqslant \sum\limits_{k = 0}^\infty {2^k a_{2^k } } .
    So if  \sum\limits_{k = 0}^\infty {2^k a_{2^k } } converges then <br />
\sum\limits_{k = 1}^\infty {a_k } converges.

    Likewise, the second inequality tells us \sum\limits_{k = 1}^\infty {2^{k - 1} a_{2^k } } \leqslant \sum\limits_{k = 2}^\infty {a_k } .
    By comparison both converge.
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