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Math Help - Need help on series! Relatively easy...

  1. #1
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    Need help on series! Relatively easy...

    Alright. We were just assigned this work and its due tomorrow. I got everything except these three problems, so help would be appreciated.

    1) Find the radius of convergence and interval of convergence of the series:

    ((-1)^n)((X^2n)/(2n)!)

    does this turn into lim as n->infinity of 2n/(2n+1) times |x^2| ?

    The ! threw me off.


    2) If R = infinity, does the series converge at only the x? For example...
    lim n-> infinity of 24n/6 |X-4| would the interval be only 4 ?

    3) This is the main problem ive been struggling on...
    Suppose that the series of CnX^n converges when x=-4 and diverges when x=6. What can be said about the convergence or divergence of the following series?

    a) Series (to infinity from n=0) of Cn,
    b) Series of Cn8^n
    c) Series of Cn(-3)^n
    d) Series of ((-1)^n)Cn9^n




    Any and all help is appreciated.
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  2. #2
    Eater of Worlds
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    Alright. We were just assigned this work and its due tomorrow. I got everything except these three problems, so help would be appreciated.

    1) Find the radius of convergence and interval of convergence of the series:

    ((-1)^n)((X^2n)/(2n)!)

    does this turn into lim as n->infinity of 2n/(2n+1) times |x^2| ?

    The ! threw me off.
    This looks like the Taylor series for cos(x).

    cos(x)=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+.......=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!}

    Compare to yours.
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by 3deltat View Post
    Alright. We were just assigned this work and its due tomorrow. I got everything except these three problems, so help would be appreciated.

    1) Find the radius of convergence and interval of convergence of the series:

    ((-1)^n)((X^2n)/(2n)!)

    does this turn into lim as n->infinity of 2n/(2n+1) times |x^2| ?

    The ! threw me off.
    Applying the ratio test gives

    \begin{aligned}\lim_{n\to\infty}\left|\frac{(-1)^{n+1}x^{2n+2}}{(2n+2)(2n+1)(2n)!}\cdot\frac{(2n  )!}{(-1)^nx^{2n}}\right|&=|x^2|\cdot\lim_{n\to\infty}\le  ft|\frac{1}{(2n+1)(2n+2)}\right|\\<br />
&=|x|\cdot{0}<1~~\forall{x}\in\mathbb{R}\end{align  ed}

    2) If R = infinity, does the series converge at only the x? For example...
    lim n-> infinity of 24n/6 |X-4| would the interval be only 4 ?
    Suppose you have a power series \sum{a_n(x-c)^n} and it converges only at its center c then the Interval of Convergence is x\in\left\{c\right\} and its radius of convergence is R=0
    3) This is the main problem ive been struggling on...
    Suppose that the series of CnX^n converges when x=-4 and diverges when x=6. What can be said about the convergence or divergence of the following series?

    a) Series (to infinity from n=0) of Cn,
    b) Series of Cn8^n
    c) Series of Cn(-3)^n
    d) Series of ((-1)^n)Cn9^n
    I suppose you mean this, \sum{c_nx^n} converges on x\in[-4,6)

    Then all your questions can be answered by evaluating of the series in question's value for x is within the interval of convergence.
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