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Math Help - Maclaurin Series questions

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    Maclaurin Series questions

    I need help with various calculus review problems. that would be great if you could.

    E) What are the first four terms of the MacClaurin series of f(x) = \frac {1}{1 = 2x}? What is its interval of convergence?

    E1) What are the first four terms of the MacClaurin series of f(x) = \frac {1}{1 = 2x}^2? What is its interval of convergence?

    E2) What are the first four terms of the MacClaurin series of ln(1 + 2x)? What is its interval of convergence?
    Last edited by mr fantastic; May 5th 2009 at 05:03 AM. Reason: Moved from original thread
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    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by corgonin View Post
    I need help with various calculus review problems. that would be great if you could.

    E) What are the first four terms of the MacClaurin series of f(x) = \frac {1}{1 = 2x}? What is its interval of convergence?

    E1) What are the first four terms of the MacClaurin series of f(x) = \frac {1}{1 = 2x}^2? What is its interval of convergence?

    E2) What are the first four terms of the MacClaurin series of ln(1 + 2x)? What is its interval of convergence?
    ... and we apply the binomial expansion...

    (1+a\cdot x)^{n}= 1 + n\cdot (ax) + \frac{n\cdot (n-1)}{2}\cdot (ax)^{2} + \dots + \frac{n\cdot (n-1) \dots (n-k+1)}{k!}\cdot (ax)^{k} + \dots (1)

    We obtain...

    \frac{1}{1+2x} = 1 - 2\cdot x + 4\cdot x^{2} - \dots + (-1)^{k}\cdot (2x)^{k} + \dots (2)

    \frac{1}{(1+2x)^{2}} = 1 - 4\cdot x + 12\cdot x^{2} - \dots + (-1)^{k}\cdot (k+1)\cdot (2x)^{k} + \dots (3)

    Because is...

    \ln (1+2x)= 2\cdot \int \frac{dx}{1+2x}

    ... from (2) ...

    \ln (1+2x) = 2\cdot x -2\cdot x^{2} + \frac{8}{3}\cdot x^{3} + \dots + (-1)^{k}\cdot \frac{2^{k+1}}{k+1}\cdot x^{k+1} + \dots (4)

    The expansion (2), (3) and (4) converge all for |x|<\frac{1}{2}...

    Kind regards

    \chi \sigma
    Last edited by mr fantastic; May 5th 2009 at 05:04 AM. Reason: Added the quote.
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