7.2E: Review of Power Series (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
Q7.1.1
1.For each power series use Theorem 7.1.3 to find the radius of convergence R. If R>0, find the open interval of convergence.
- ∞∑n=0(−1)n2nn(x−1)n
- ∞∑n=02nn(x−2)n
- ∞∑n=0n!9nxn
- ∞∑n=0n(n+1)16n(x−2)n
- ∞∑n=0(−1)n7nn!xn
- ∞∑n=03n4n+1(n+1)2(x+7)n
2. Suppose there’s an integer M such that bm≠0 for m≥M, and limm→∞|bm+1bm|=L, where 0≤L≤∞. Show that the radius of convergence of ∞∑m=0bm(x−x0)2m is R=1/√L, which is interpreted to mean that R=0 if L=∞ or R=∞ if L=0.
3. For each power series, use the result of Exercise 7.1.2 to find the radius of convergence R. If R>0, find the open interval of convergence.
- ∞∑m=0(−1)m(3m+1)(x−1)2m+1
- ∞∑m=0(−1)mm(2m+1)2m(x+2)2m
- ∞∑m=0m!(2m)!(x−1)2m
- ∞∑m=0(−1)mm!9m(x+8)2m
- ∞∑m=0(−1)m(2m−1)3mx2m+1
- ∞∑m=0(x−1)2m
4. Suppose there’s an integer M such that bm≠0 for m≥M, and limm→∞|bm+1bm|=L, where 0≤L≤∞. Let k be a positive integer. Show that the radius of convergence of ∞∑m=0bm(x−x0)km is R=1/k√L, which is interpreted to mean that R=0 if L=∞ or R=∞ if L=0.
5. For each power series use the result of Exercise 7.1.4 to find the radius of convergence R. If R>0, find the open interval of convergence.
- ∞∑m=0(−1)m(27)m(x−3)3m+2
- ∞∑m=0x7m+6m
- ∞∑m=09m(m+1)(m+2)(x−3)4m+2
- ∞∑m=0(−1)m2mm!x4m+3
- ∞∑m=0m!(26)m(x+1)4m+3
- ∞∑m=0(−1)m8mm(m+1)(x−1)3m+1
6. Graph y=sinx and the Taylor polynomial T2M+1(x)=M∑n=0(−1)nx2n+1(2n+1)! on the interval (−2π,2π) for M=1, 2, 3, …, until you find a value of M for which there’s no perceptible difference between the two graphs.
7. Graph y=cosx and the Taylor polynomial T2M(x)=M∑n=0(−1)nx2n(2n)! on the interval (−2π,2π) for M=1, 2, 3, …, until you find a value of M for which there’s no perceptible difference between the two graphs.
8. Graph y=1/(1−x) and the Taylor polynomial TN(x)=N∑n=0xn on the interval [0,.95] for N=1, 2, 3, …, until you find a value of N for which there’s no perceptible difference between the two graphs. Choose the scale on the y-axis so that 0≤y≤20.
9. Graph y=coshx and the Taylor polynomial T2M(x)=M∑n=0x2n(2n)! on the interval (−5,5) for M=1, 2, 3, …, until you find a value of M for which there’s no perceptible difference between the two graphs. Choose the scale on the y-axis so that 0≤y≤75.
10. Graph y=sinhx and the Taylor polynomial T2M+1(x)=M∑n=0x2n+1(2n+1)! on the interval (−5,5) for M=0, 1, 2, …, until you find a value of M for which there’s no perceptible difference between the two graphs. Choose the scale on the y-axis so that −75 ≤ y≤ 75.
Q7.1.2
In Exercises 7.1.11-7.1.15 find a power series solution \(y(x)=\sum_{n=0}^{\infty} a_{n}x^{n} \nonumber \].
11. (2+x)y″
12. (1+3x^2)y''+3x^2y'-2y
13. (1+2x^2)y''+(2-3x)y'+4y
14. (1+x^2)y''+(2-x)y'+3y
15. (1+3x^2)y''-2xy'+4y
Q7.1.3
16. 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
17. 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
18. Do the following experiment for various choices of real numbers a_0 and a_1.
- Use differential equations software to solve the initial value problem (2-x)y''+2y=0,\quad y(0)=a_0,\quad y'(0)=a_1,\nonumber numerically on (-1.95,1.95). Choose the most accurate method your software package provides. (See Section 10.1 for a brief discussion of one such method.)
- For N=2, 3, 4, …, compute a_2, …, a_N from Equation 7.1.18 and graph T_N(x)=\displaystyle \sum_{n=0}^N a_nx^n\nonumber and the solution obtained in (a) on the same axes. Continue increasing N until it is obvious that there’s no point in continuing. (This sounds vague, but you’ll know when to stop.)
19. Follow the directions of Exercise 7.1.18 for the initial value problem (1+x)y''+2(x-1)^2y'+3y=0,\quad y(1)=a_0,\quad y'(1)=a_1,\nonumber on the interval (0,2). Use Equations 7.1.24 and 7.1.25 to compute \{a_n\}.
20. Suppose the series \displaystyle \sum_{n=0}^\infty a_nx^n converges on an open interval (-R,R), let r be an arbitrary real number, and define y(x)=x^r\displaystyle \sum_{n=0}^\infty a_nx^n=\displaystyle \sum_{n=0}^\infty a_nx^{n+r}\nonumber on (0,R). Use Theorem 7.1.4 and the rule for differentiating the product of two functions to show that \begin{aligned} y'(x)&={\displaystyle \sum_{n=0}^\infty (n+r)a_nx^{n+r-1}},\\[4pt] y''(x)&={\displaystyle \sum_{n=0}^\infty(n+r)(n+r-1)a_nx^{n+r-2}},\\[4pt] &\vdots&\\[4pt] y^{(k)}(x)&={\displaystyle \sum_{n=0}^\infty(n+r)(n+r-1)\cdots(n+r-k)a_nx^{n+r-k}}\end{aligned}\nonumber on (0,R)
Q7.1.4
21. x^2(1-x)y''+x(4+x)y'+(2-x)y
22. x^2(1+x)y''+x(1+2x)y'-(4+6x)y
23. x^2(1+x)y''-x(1-6x-x^2)y'+(1+6x+x^2)y
24. x^2(1+3x)y''+x(2+12x+x^2)y'+2x(3+x)y
25. x^2(1+2x^2)y''+x(4+2x^2)y'+2(1-x^2)y
26. x^2(2+x^2)y''+2x(5+x^2)y'+2(3-x^2)y