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Section 12.1 Assignment 1

  1. Test the following series for convergence. (24)

    1. \begin{equation*} \sum_{n=0}^\infty \frac{n^2 + 1}{n^3 - 4} \end{equation*}

    2. \begin{equation*} \sum_{n=0}^\infty \left( \sqrt{\frac{1}{2}} \right)^{2n} \end{equation*}

    3. \begin{equation*} \sum_{n=0}^\infty \left( \frac{1}{\pi - 3} \right)^n \end{equation*}

    4. \begin{equation*} \sum_{n=0}^\infty \frac{(n+1)! (2n)!}{(n!)^3} \end{equation*}

    5. \begin{equation*} \sum_{n=2}^\infty \frac{\ln n}{n^2} \end{equation*}

    6. \begin{equation*} \sum_{n=1}^\infty \frac{n^2 + n^3(\sin (n \pi))}{n^4 + 4} \end{equation*}

    7. \begin{equation*} \sum_{n=1}^\infty \frac{(\ln n)^2}{n^2 + 1} \end{equation*}

    8. \begin{equation*} \sum_{n=1}^\infty \frac{(-1)^n n^2}{n^2 + 4} \end{equation*}

  2. Calculate the Taylor series for each function, if possible, around the given centre point \(\alpha\) and determine the radius of convergence. You can use commonly known series and adapt them, or calculate directly using derivatives to determine the series. (20)

    1. \begin{align*} \amp \sin (6x^2) \amp \amp \alpha = 0 \end{align*}

    2. \begin{align*} \amp \frac{1}{1 - x^4} \amp \amp \alpha = 1 \end{align*}

    3. \begin{align*} \amp \ln x \amp \amp \alpha = e \end{align*}

  3. Use the Lagrange error bounds to solve the following problems. (12)

    1. Given \(\sin x\) as a Taylor series centered at \(0\text{,}\) what interval allows the seventh order approximation to have accuracy \(\frac{1}{1000}\text{.}\) (You may approximate the interval to the nearest hundredth.)

    2. Given \(\ln x\) as a Taylor series centered at \(e\text{,}\) what accuracy is guaranteed by using a tenth order approximation on the interval \((e-1,e+1)\text{?}\)

  4. Given \(\frac{1}{1-x}\) as a Taylor series centered at \(0\text{,}\) show that the Lagrange error calculation cannot determine the order of approximation required to ensure that the error is less than \(\frac{1}{1000}\) on the interval \(\left( \frac{-1}{2}, \frac{1}{2} \right)\text{?}\)

  5. Let \(f\) and \(g\) be positive continuous functions which both grow to inifnnity as \(x \rightarrow \infty\text{.}\) Assume that the following series has a finite, non-zero radius of convergence \(R\text{.}\) (10)

    \begin{equation*} \sum_{n=0}^\infty \frac{f(n)}{g(n)} x^n \end{equation*}

    What happens to \(R\text{,}\) if anything, under the following changes. Be as specific as you can and give reasons.

    1. \(f\) is replace by \(2f\text{.}\)

    2. \(g\) is replace by \(\frac{g}{3}\text{.}\)

    3. \(f\) is replace by \(f^2\text{.}\)

    4. \(g\) is replace by \(g^3\text{.}\)

    5. \(g\) is replace by \(g+1\text{.}\)