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Mathematics LibreTexts

Chapter 10 Review Exercises

  • Page ID
    10803
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    True or False? In exercises 1 - 4, 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 \(5\), then the radius of convergence for the series \(\displaystyle \sum_{n=1}^∞na_nx^{n−1}\) is also \(5\).

    Answer:
    True

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

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

    Answer:
    True

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

     

    In exercises 5 - 8, find the radius of convergence and the interval of convergence for the given series.

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

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

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

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

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

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

     

    In exercises 9 - 10, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

    9) \(f(x)=\dfrac{x^2}{x+3}\)

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

    10) \(f(x)=\dfrac{8x+2}{2x^2−3x+1}\)

     

    In exercises 11 - 12, find the power series for the given function using term-by-term differentiation or integration.

    11) \(f(x)=\tan^{−1}(2x)\)

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

    12) \(f(x)=\dfrac{x}{(2+x^2)^2}\)

     

    In exercises 13 - 14, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

    13) \(f(x)=x^3−2x^2+4, \quad a=−3\)

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

    14) \(f(x)=e^{1/(4x)}, \quad a=4\)

     

    In exercises 15 - 16, find the Maclaurin series for the given function.

    15) \(f(x)=\cos(3x)\)

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

    16) \(f(x)=\ln(x+1)\)

     

    In exercises 17 - 18, find the Taylor series at the given value.

    17) \(f(x)=\sin x, \quad a=\frac{π}{2}\)

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

    18) \(f(x)=\dfrac{3}{x},\quad a=1\)

     

    In exercises 19 - 20, find the Maclaurin series for the given function.

    19) \(f(x)=e^{−x^2}−1\)

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

    20) \(f(x)=\cos x−x\sin x\)

     

    In exercises 21 - 23, find the Maclaurin series for \(F(x)=∫^x_0f(t)dt\) by integrating the Maclaurin series of \(f(x)\) term by term.

    21) \(f(x)=\dfrac{\sin x}{x}\)

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

    22) \(f(x)=1−e^x\)

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

    Answer:
    Answers may vary.

     

    Exercises 24 - 26 consider problems of annuity payments.

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

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

    Answer:
    \(2.5\%\)

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

     

    Contributors

    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

     

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