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Math Help - radius of convergence

  1. #1
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    Question radius of convergence

    Hi,
    Could someone help me with these questions,I dont understand how you can find the radius of convergence:

    1.f(x)=(1+4x)^3/2

    State the radius of convergence of this power series

    (I also had to use the General Binomial Theorem to determine the first four terms of the Taylor series at 0 for this series but I have done this)

    2.g(x)=6(1+4x)^1/2

    State the radius of convergence of this power series

    (I also had to write down the first three terms of the Tayor series at 0 but I have done this)
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  2. #2
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    Quote Originally Posted by mountaindream View Post
    Hi,
    Could someone help me with these questions,I dont understand how you can find the radius of convergence:

    1.f(x)=(1+4x)^3/2

    State the radius of convergence of this power series

    (I also had to use the General Binomial Theorem to determine the first four terms of the Taylor series at 0 for this series but I have done this)

    2.g(x)=6(1+4x)^1/2

    State the radius of convergence of this power series

    (I also had to write down the first three terms of the Tayor series at 0 but I have done this)
    The first thing you need to do is get the general term of the power series for each of the given functions.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    The first thing you need to do is get the general term of the power series for each of the given functions.
    I dont understand what you mean-do you know how to do this?
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  4. #4
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    Quote Originally Posted by mountaindream View Post
    I dont understand what you mean-do you know how to do this?
    Have you studied power series at all (eg. Maclaurin series, etc.)? Have you learned how to write functions in the form f(x) = \sum_{n=1}^{\infty} a_n x^n?

    The functions f(x) and g(x) are not power series. So the first step is to write them in the form f(x) = \sum_{n=1}^{\infty} a_n x^n.
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    Quote Originally Posted by mr fantastic View Post
    Have you studied power series at all (eg. Maclaurin series, etc.)? Have you learned how to write functions in the form f(x) = \sum_{n=1}^{\infty} a_n x^n?

    The functions f(x) and g(x) are not power series. So the first step is to write them in the form f(x) = \sum_{n=1}^{\infty} a_n x^n.
    No I have not learnt about that kind of series yet. ~It just seems it should be obvious because the question says "state" which implies it should be simple to detect the radius of convergence just by looking at the question. Thank-you for your replies Any more hinters as altough I'm trying my best I'm still not getting it . x
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  6. #6
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    Quote Originally Posted by mountaindream View Post
    No I have not learnt about that kind of series yet.
    What do you know about power series?

    ~It just seems it should be obvious because the question says "state" which implies it should be simple to detect the radius of convergence just by looking at the question. Thank-you for your replies Any more hinters as altough I'm trying my best I'm still not getting it . x
    No its not. If you expand a function as a power series about some point, the radius of convergence depends on which point you choose.

    Now if you specify that the expansion is about x=0 then you might get an answer.

    RonL
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  7. #7
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    Quote Originally Posted by CaptainBlack View Post
    What do you know about power series?



    No its not. If you expand a function as a power series about some point, the radius of convergence depends on which point you choose.

    Now if you specify that the expansion is about x=0 then you might get an answer.

    RonL
    ok for the first expansion I got: 1+6x+6x^2-4x^3
    for 2.I got: -root6+1/root12x-1/root48x^2

    Could you tell me where to go from here to state the radius of convergence? I'm struggling with the concept as I dont understand it. I only have a text book explanatin which is very brief!
    Responses greatly appreciated
    x
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  8. #8
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    Quote Originally Posted by mountaindream View Post
    No I have not learnt about that kind of series yet. ~It just seems it should be obvious because the question says "state" which implies it should be simple to detect the radius of convergence just by looking at the question. Thank-you for your replies Any more hinters as altough I'm trying my best I'm still not getting it . x
    Are you doing binomial expansions?

    Then the first is:

    f(x)=1+\left( 3/2 \right) (4x) + \frac{\left( (3/2)(3/2-1) \right)}{2!} (4x)^2+ \frac{\left( (3/2)(3/2-1)(3/2-2) \right)}{3!} (4x)^3+...

    which needs a bit of simplifying

    RonL
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  9. #9
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    Quote Originally Posted by CaptainBlack View Post
    Are you doing binomial expansions?

    Then the first is:

    f(x)=1+\left( 3/2 \right) (4x) + \frac{\left( (3/2)(3/2-1) \right)}{2!} (4x)^2+ \frac{\left( (3/2)(3/2-1)(3/2-2) \right)}{3!} (4x)^3+...

    which needs a bit of simplifying

    RonL
    yes I got that result and then simplified it to get the polynomail whihc I stated previously but I dont understand how to find the radius of convergence from that :S
    Thankyou x
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  10. #10
    Grand Panjandrum
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    The radius of convergence of the binomial expansion of (1+x)^a, where a is not a non-negative integer is 1.

    RonL
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