8.2.1: Review of Power Series (Exercises)
- Page ID
- 103525
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Exercises 1-18 use Theorem 8.2.2 (the ratio test) to find the radius of convergence \(R\). If \(R>0\), find the open interval of convergence.
1. \({\displaystyle \sum_{n=0}^\infty {(-1)^n\over2^nn}(x-1)^n}\)
2. \({\displaystyle \sum_{n=0}^\infty 2^nn(x-2)^n}\)
3. \({\displaystyle \sum_{n=0}^\infty {n!\over9^n}x^n}\)
4. \({\displaystyle \sum_{n=0}^\infty{n(n+1)\over16^n}(x-2)^n}\)
5. \({\displaystyle \sum_{n=0}^\infty (-1)^n{7^n\over n!}x^n}\)
6. \({\displaystyle \sum_{n=0}^\infty {3^n\over4^{n+1}(n+1)^2}(x+7)^n}\)
7. \({\displaystyle \sum_{m=0}^\infty (-1)^m(3m+1)(x-1)^{2m+1}}\)
8. \({\displaystyle \sum_{m=0}^\infty (-1)^m{m(2m+1)\over2^m}(x+2)^{2m}}\)
9. \({\displaystyle \sum_{m=0}^\infty {m!\over(2m)!}(x-1)^{2m}}\)
10. \({\displaystyle \sum_{m=0}^\infty (-1)^m{m!\over9^m}(x+8)^{2m}}\)
11. \({\displaystyle \sum_{m=0}^\infty(-1)^m{(2m-1)\over3^m}x^{2m+1}}\)
12. \({\displaystyle \sum_{m=0}^\infty(x-1)^{2m}}\)
13. \({\displaystyle \sum_{m=0}^\infty{(-1)^m\over(27)^m}(x-3)^{3m+2}}\)
14. \({\displaystyle \sum_{m=0}^\infty{x^{7m+6}\over m}}\)
15. \({\displaystyle \sum_{m=0}^\infty{9^m(m+1)\over(m+2)}(x-3)^{4m+2}}\)
16. \({\displaystyle \sum_{m=0}^\infty(-1)^m{2^m\over m!}x^{4m+3}}\)
17. \({\displaystyle \sum_{m=0}^\infty{m!\over(26)^m}(x+1)^{4m+3}}\)
18. \({\displaystyle \sum_{m=0}^\infty{(-1)^m\over8^mm(m+1)}(x-1)^{3m+1}}\)
In Exercises 19-23 express as a power series in \(x\).
19. \((2+x)y''+xy'+3y\)
20. \((1+3x^2)y''+3x^2y'-2y\)
21. \((1+2x^2)y''+(2-3x)y'+4y\)
22. \((1+x^2)y''+(2-x)y'+3y\)
23. \((1+3x^2)y''-2xy'+4y\)
24. Suppose \(y(x)=\displaystyle \sum_{n=0}^\infty a_n(x+1)^n\) on an open interval that contains \(x_0=-1\). Find a power series in \(x+1\) for \[xy''+(4+2x)y'+(2+x)y.\nonumber \]
25. Suppose \(y(x)=\displaystyle \sum_{n=0}^\infty a_n(x-2)^n\) on an open interval that contains \(x_0=2\). Find a power series in \(x-2\) for \[x^2y''+2xy'-3xy.\nonumber \]