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Math Help - Radius of convergence and interval of convergence

  1. #1
    Senior Member Educated's Avatar
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    Radius of convergence and interval of convergence

    (a) Write down the Maclaurin series for \dfrac{1}{1-x}. What it it's radius of convergence?


    (b) Write down the Maclaurin series for \dfrac{1}{1+x}


    (c) Use (b) to find the Maclaurin series for f(x) = ln (1+x)

    (d) What is the radius of convergence found in (c)?

    (e) What is the interval of convergence found in (c)?
    I can do a, b, c and d alright, but I'm stuck on e.

    (a) \dfrac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + x^5 ...
    The radius of convergence is 1 (It's a geometric series where x is the ratio).

    (b) \dfrac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 ...

    (c) \displaystyle \int \dfrac{1}{1+x} = \ln (1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \dfrac{x^5}{5} - \dfrac{x^6}{6} ...

    (d) The radius of convergence is still 1, integrating does not alter the radius of convergence.

    Now how do I go about checking the interval of convergence? Is it just (-1, 1)?
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  2. #2
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    Re: Radius of convergence and interval of convergence

    Have you tried substituting in x = -1 and x = 1? Obviously the series must be divergent when x = -1 because the logarithm is undefined at 0. As for when x = 1, we then have \displaystyle \begin{align*} 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots - \dots \end{align*}, which is the alternating harmonic series and is known to be convergent.
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  3. #3
    Senior Member Educated's Avatar
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    Re: Radius of convergence and interval of convergence

    Oh right, haha, so the interval of convergence is (-1, 1]

    How would I know that the interval of convergence is around 0? Is there any series (can you give an example?) where the interval of convergence is centered around something else?
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    Re: Radius of convergence and interval of convergence

    A MacLaurin Series is centred at 0 and so therefore is its radius of convergence, a general Taylor series centred at x = c will have a radius of convergence centred at x = c.
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