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11.13: Additional Exercises

  • Page ID
    149554
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    These problems require the techniques of this chapter, and are in no particular order. Some problems may be done in more than one way.

    Exercises \(\PageIndex{}\)

    Exercises \(\PageIndex{1}\) to \(\PageIndex{17}\), determine whether the series converges.

    Exercise \(\PageIndex{1}\)

    \(\displaystyle \sum_{n=0}^{\infty}\dfrac{n}{n^2+4}\)

    Answer

    Diverges

    Exercise \(\PageIndex{2}\)

    \(\displaystyle \dfrac{1}{1\cdot 2}+ \dfrac{1}{3\cdot 4}+ \dfrac{1}{5\cdot 6}+ \dfrac{1}{7\cdot 8}+ \cdots\)

    Answer

    Converges

    Exercise \(\PageIndex{3}\)

    \(\displaystyle \sum_{n=0}^{\infty}\dfrac{n}{(n^2+4)^2}\)

    Answer

    Converges

    Exercise \(\PageIndex{4}\)

    \(\displaystyle\sum_{n=0}^\infty \dfrac{n!}{8^n}\)

    Answer

    Diverges

    Exercise \(\PageIndex{5}\)

    \(\displaystyle 1- \dfrac{3}{4}+ \dfrac{5}{8}- \dfrac{7}{12}+ \dfrac{9}{16}+ \cdots\)

    Answer

    Diverges

    Exercise \(\PageIndex{6}\)

    \(\displaystyle\sum_{n=0}^\infty \dfrac{1}{\sqrt{n^2+4}}\)

    Answer

    Diverges

    Exercise \(\PageIndex{7}\)

    \(\displaystyle\sum_{n=0}^\infty \dfrac{\sin^3(n)}{n^2}\)

    Answer

    Converges

    Exercise \(\PageIndex{8}\)

    \(\displaystyle\sum_{n=0}^{\infty} \dfrac{n}{e^n}\)

    Answer

    Converges

    Exercise \(\PageIndex{9}\)

    \(\displaystyle\sum_{n=0}^\infty \dfrac{n!}{1\cdot 3\cdot 5\cdots(2n-1)}\)

    Answer

    Converges

    Exercise \(\PageIndex{10}\)

    \(\displaystyle\sum_{n=1}^\infty \dfrac{1}{n\sqrt{n}}\)

    Answer

    Converges

    Exercise \(\PageIndex{11}\)

    \(\displaystyle \dfrac{1}{2\cdot 3\cdot 4}+ \dfrac{2}{3\cdot 4\cdot 5}+ \dfrac{3}{4\cdot 5\cdot 6}+ \dfrac{4}{5\cdot 6 \cdot 7}+ \cdots\)

    Answer

    Converges

    Exercise \(\PageIndex{12}\)

    \(\displaystyle\sum_{n=1}^\infty \dfrac{1\cdot 3\cdot 5\cdots(2n-1)}{(2n)!}\)

    Answer

    Converges

    Exercise \(\PageIndex{13}\)

    \(\displaystyle\sum_{n=0}^\infty \dfrac{6^n}{n!}\)

    Answer

    Converges

    Exercise \(\PageIndex{14}\)

    \(\displaystyle\sum_{n=1}^\infty \dfrac{(-1)^{n-1}}{\sqrt{n}}\)

    Answer

    Converges

    Exercise \(\PageIndex{15}\)

    \(\displaystyle\sum_{n=1}^\infty \dfrac{2^n 3^{n-1}}{n!}\)

    Answer

    Converges

    Exercise \(\PageIndex{16}\)

    \(\displaystyle 1+ \dfrac{5^2}{2^2}+ \dfrac{5^4}{(2\cdot 4)^2}+ \dfrac{5^6}{(2\cdot 4\cdot 6)^2}+ \dfrac{5^8}{(2\cdot 4\cdot 6\cdot 8)^2}+ \cdots\)

    Answer

    Converges

    Exercise \(\PageIndex{17}\)

    \(\displaystyle\sum_{n=1}^\infty \sin(1/n)\)

    Answer

    Diverges

    Exercises \(\PageIndex{18}\) to \(\PageIndex{24}\), find the interval and radius of convergence. You need not check the endpoints of the intervals.

    Exercise \(\PageIndex{18}\)

    \(\displaystyle\sum_{n=0}^\infty \dfrac{2^n}{n!}x^n\)

    Answer

    \((-\infty,\infty)\)

    Exercise \(\PageIndex{19}\)

    \(\displaystyle \sum_{n=0}^\infty \dfrac{x^n}{1+3^n}\)

    Answer

    \((-3,3)\)

    Exercise \(\PageIndex{20}\)

    \(\displaystyle\sum_{n=1}^\infty \dfrac{x^n}{n3^n}\)

    Answer

    \((-3,3)\)

    Exercise \(\PageIndex{21}\)

    \(\displaystyle x+ \dfrac{1}{2} \dfrac{x^3}{3} + \dfrac{1\cdot 3}{2\cdot 4} \dfrac{x^5}{5}+ \dfrac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6} \dfrac{x^7}{7}+\cdots\)

    Answer

    \((-1,1)\)

    Exercise \(\PageIndex{22}\)

    \(\displaystyle\sum_{n=1}^\infty \dfrac{n!}{n^2} x^n\)

    Answer

    Radius is \(0\) - it converges only when \(x=0\).

    Exercise \(\PageIndex{23}\)

    \(\displaystyle\sum_{n=1}^\infty \dfrac{(-1)^n}{n^2 3^n} x^{2n}\)

    Answer

    \((-\sqrt{3},\sqrt{3})\)

    Exercise \(\PageIndex{24}\)

    \(\displaystyle\sum_{n=0}^\infty \dfrac{(x-1)^n}{n!}\)

    Answer

    \((-\infty,\infty)\)

    Exercises \(\PageIndex{25}\) to \(\PageIndex{30}\), find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

    Exercise \(\PageIndex{25}\)

    \(2^x\)

    Answer

    \(\displaystyle \sum_{n=0}^{\infty}\dfrac{\big(\ln(2)\big)^n}{n!}x^n\)

    Exercise \(\PageIndex{26}\)

    \(\ln(1+x)\)

    Answer

    \(\displaystyle \sum_{n=0}^{\infty}\dfrac{(-1)^n}{n+1}x^{n+1}\)

    Exercise \(\PageIndex{27}\)

    \(\ln\left(\dfrac{1+x}{1-x}\right)\)

    Answer

    \(\displaystyle \sum_{n=0}^{\infty}\dfrac{2}{2n+1}x^{2n+1}\)

    Exercise \(\PageIndex{28}\)

    \(\sqrt{1+x}\)

    Answer

    \(\displaystyle 1+\dfrac{x}{2}+\sum_{n=2}^{\infty}(-1)^{n+1}\dfrac{1\cdot 3\cdot 5\cdots(2n-3)}{2^{n}n!}x^n\)

    Exercise \(\PageIndex{29}\)

    \(\dfrac{1}{1+x^2}\)

    Answer

    \(\displaystyle \sum_{n=0}^{\infty}(-1)^{n}x^{2n}\)

    Exercise \(\PageIndex{30}\)

    \(\arctan(x)\)

    Answer

    \(\displaystyle \sum_{n=0}^{\infty}\dfrac{(-1)^n}{2n+1}x^{2n+1}\)

    Exercise \(\PageIndex{31}\)

    Use the answer to the previous problem to discover a series for a well-known mathematical constant.

    Answer

    \(\displaystyle \pi=\sum_{n=0}^{\infty}(-1)^n\dfrac{4}{2n+1}\)


    This page titled 11.13: Additional Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David Guichard via source content that was edited to the style and standards of the LibreTexts platform.