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Mathematics LibreTexts

1 E: Chapter Exercises

  • Page ID
    17138
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    Exercise \(\PageIndex{1}\)

    True or False? In the following exercises, justify your answer with a proof or a counterexample.

    1. If the radius of convergence for a power series \(\displaystyle \sum_{n=0}^∞a_nx^n\) is \(\displaystyle 5\), then the radius of convergence for the series \(\displaystyle \sum_{n=1}^∞na_nx^{n−1}\) is also \(\displaystyle 5\).

    Answer

    True

    2. Power series can be used to show that the derivative of \(\displaystyle e^x\) is \(\displaystyle e^x\). (Hint: Recall that \(\displaystyle e^x=\sum_{n=0}^∞\frac{1}{n!}x^n.\))

    3. For small values of \(\displaystyle x,sinx≈x.\)

    Answer

    True

    4. The radius of convergence for the Maclaurin series of \(\displaystyle f(x)=3^x\) is \(\displaystyle 3\)

    Exercise \(\PageIndex{2}\)

    In the following exercises, find the radius of convergence and the interval of convergence for the given series.

    1. \(\displaystyle \sum_{n=0}^∞n^2(x−1)^n\)

    Answer

    ROC: \(\displaystyle 1\); IOC: \(\displaystyle (0,2)\)

    2. \(\displaystyle \sum_{n=0}^∞\frac{x^n}{n^n}\)

    3. \(\displaystyle \sum_{n=0}^∞\frac{3nx^n}{12^n}\)

    Answer

    ROC: \(\displaystyle 12;\) IOC: \(\displaystyle (−16,8)\)

    4. \(\displaystyle \sum_{n=0}^∞\frac{2^n}{e^n}(x−e)^n\)

    Exercise \(\PageIndex{3}\)

    In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

    1. \(\displaystyle f(x)=\frac{x^2}{x+3}\)

    Answer

    \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{3^{n+1}}x^n;\) ROC: \(\displaystyle 3\); IOC: \(\displaystyle (−3,3)\)

    2. \(\displaystyle f(x)=\frac{8x+2}{2x^2−3x+1}\)

    Exercise \(\PageIndex{4}\)

    In the following exercises, find the power series for the given function using term-by-term differentiation or integration.

    1. \(\displaystyle f(x)=tan^{−1}(2x)\)

    Answer

    integration: \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{2n+1}(2x)^{2n+1}\)

    2. \(\displaystyle f(x)=\frac{x}{(2+x^2)^2}\)

    Exercise \(\PageIndex{5}\)

    In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

    1. \(\displaystyle f(x)=x^3−2x^2+4,a=−3\)

    Answer

    \(\displaystyle p_4(x)=(x+3)^3−11(x+3)^2+39(x+3)−41;\) exact

    2. \(\displaystyle f(x)=e^{1/(4x)},a=4\)

    Exercise \(\PageIndex{6}\)

    In the following exercises, find the Maclaurin series for the given function.

    1. \(\displaystyle f(x)=cos(3x)\)

    Answer

    \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n(3x)^{2n}}{2n!}\)

    2. \(\displaystyle f(x)=ln(x+1)\)

    Exercise \(\PageIndex{7}\)

    In the following exercises, find the Taylor series at the given value.

    1. \(\displaystyle f(x)=sinx,a=\frac{π}{2}\)

    Answer

    \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{(2n)!}(x−\frac{π}{2})^{2n}\)

    2. \(\displaystyle f(x)=\frac{3}{x},a=1\)

    Exercise \(\PageIndex{8}\)

    In the following exercises, find the Maclaurin series for the given function.

    1. \(\displaystyle f(x)=e^{−x^2}−1\)

    Answer

    \(\displaystyle \sum_{n=1}^∞\frac{(−1)^n}{n!}x^{2n}\)

    2. \(\displaystyle f(x)=cosx−xsinx\)

    Exercise \(\PageIndex{9}\)

    In the following exercises, find the Maclaurin series for \(\displaystyle F(x)=∫^x_0f(t)dt\) by integrating the Maclaurin series of \(\displaystyle f(x)\) term by term.

    1. \(\displaystyle f(x)=\frac{sinx}{x}\)

    Answer

    \(\displaystyle F(x)=\sum_{n=0}^∞\frac{(−1)^n}{(2n+1)(2n+1)!}x^{2n+1}\)

    2. \(\displaystyle f(x)=1−e^x\)

    3. Use power series to prove Euler’s formula: \(\displaystyle e^{ix}=cosx+isinx\)

    Exercise \(\PageIndex{10}\)

    The following exercises consider problems of annuity payments.

    1. For annuities with a present value of \(\displaystyle $1\) million, calculate the annual payouts given over \(\displaystyle 25\) years assuming interest rates of \(\displaystyle 1%,5%\), and \(\displaystyle 10%.\)

    2. A lottery winner has an annuity that has a present value of \(\displaystyle $10\) million. What interest rate would they need to live on perpetual annual payments of \(\displaystyle $250,000\)?

    Answer

    \(\displaystyle 2.5%\)

    3. Calculate the necessary present value of an annuity in order to support annual payouts of \(\displaystyle $15,000\) given over \(\displaystyle 25\) years assuming interest rates of \(\displaystyle 1%,5%\),and \(\displaystyle 10%.\)