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Math Help - Help needed on two questions!

  1. #1
    Junior Member
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    Help needed on two questions!

    Hi, i have recently been trying to complete an assignment of 10 mathematical questions. I have managed to answer 8 of the 10 questions but unfortunately have been unable to answer two. Can some please show me how to calculate the following as I have no idea how to attempt them. Below are the questions:

    1) Given that the Maclaurin expansion of 1/1+x = 1 - x + x^2 - x^3... deduce the expansion of 1/1+cos(x) as far as the term in x^3.

    2) If a>1 and n^√a = 1+x , prove that 0 < x < a/n. Deduce that n^√a --> 1 and n --> infiniti (∞). What is the corresponding result if 0 < a < 1?

    Can anyone show me a step by step solution so i get a better understanding on how to answer these questions.

    Thanks
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  2. #2
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    Quote Originally Posted by NFS1 View Post
    1) Given that the Maclaurin expansion of 1/(1+x) = 1 - x + x^2 - x^3... deduce the expansion of 1/(1+cos(x)) as far as the term in x^3.

    So, \displaystyle {{1}\over{1+\cos(x)}}=1-\cos(x)+\cos^2(x)-\cos^3(x)+\dots

    Do you know the Maclaurin expansion of \displaystyle \cos(x) ?

    Plug that in for each cosine in your expansion & do some algebra.
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  3. #3
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    Is "n^√a" supposed to be n^{\sqrt{a}} or \sqrt[n]{a}? Assuming that it is the second, it is easy to see that x= \sqrt[n]{a}- 1> 0. You can then expand (1+ x)^n= a using the binomial theorem to show that a= 1+ nx+ higher power terms so that a/n< x+ positive terms.
    Last edited by HallsofIvy; December 12th 2010 at 10:50 AM.
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