3.1E: Exercises
This page is a draft and is under active development.
( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercise 3.1E.1
For each power series use Theorem (3.1.3) to find the radius of convergence R. If R>0, find the open interval of convergence.
(a) ∞∑n=0(−1)n2nn(x−1)n
(b) ∞∑n=02nn(x−2)n
(c) ∞∑n=0n!9nxn
(d) ∞∑n=0n(n+1)16n(x−2)n
(e) ∞∑n=0(−1)n7nn!xn
(f) ∞∑n=03n4n+1(n+1)2(x+7)n
Exercise 3.1E.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.
Hint: Apply Theorem (3.1.3) to the series ∑∞m=0bmzm and then let z=(x−x0)2.
- Answer
-
From Theorem~3.1.3, ∑∞m=0bmzm converges
if |z|<1/L and diverges if |z|>1/L .
Therefore,
∑∞m=0bm(x−x0)2 converges if |x−x0|<1/√L
and diverges if |x−x0|>1/√L .
Exercise 3.1E.3
For each power series, use the result of Exercise (3.1E.2) to find the radius of convergence R. If R>0, find the open interval of convergence.
(a) ∞∑m=0(−1)m(3m+1)(x−1)2m+1
(b) ∞∑m=0(−1)mm(2m+1)2m(x+2)2m
(c) ∞∑m=0m!(2m)!(x−1)2m
(d) ∞∑m=0(−1)mm!9m(x+8)2m
(e) ∞∑m=0(−1)m(2m−1)3mx2m+1
(f) ∞∑m=0(x−1)2m
Exercise 3.1E.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.
Hint: Apply Theorem (3.1.3) to the series ∑∞m=0bmzm and then let z=(x−x0)k.
- Answer
-
From Theorem~3.1.3, ∑∞m=0bmzm converges
if |z|<1/L and diverges if |z|>1/L .
Therefore,
∑∞m=0bm(x−x0)km converges if
|x−x0|<1/k√L and diverges if |x−x0|>1/k√L .
Exercise 3.1E.5
For each power series use the result of Exercise (3.1E.4) to find the radius of convergence R. If R>0, find the open interval of convergence.
(a) ∞∑m=0(−1)m(27)m(x−3)3m+2
(b) ∞∑m=0x7m+6m
(c) ∞∑m=09m(m+1)(m+2)(x−3)4m+2
(d) ∞∑m=0(−1)m2mm!x4m+3
(e) ∞∑m=0m!(26)m(x+1)4m+3
(f) ∞∑m=0(−1)m8mm(m+1)(x−1)3m+1
Exercise 3.1E.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.
Exercise 3.1E.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.
Exercise 3.1E.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.
Exercise 3.1E.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.
Exercise 3.1E.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.
In Exercises (3.1E.11) to (3.1E.15), find a power series solution y(x)=∑∞n=0anxn.
Exercise 3.1E.11
(2+x)y″
Exercise \PageIndex{12}
(1+3x^2)y''+3x^2y'-2y
- Answer
-
(1+3x^2)y''+3x^2y'-2y= \sum_{n=2}^\infty n(n-1)a_nx^{n-2} +3\sum_{n=2}^\infty n(n-1)a_nx^n +3\sum_{n=1}^\infty na_nx^{n+1}-2\sum_{n=0}^\infty a_nx^n =\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n +3\sum_{n=1}^\infty n(n-1)na_nx^n +3\sum_{n=1}^\infty (n-1)a_{n-1}x^n-2\sum_{n=0}^\infty a_nx^n =2a_2-2a_0+\sum_{n=1}^\infty [(n+2)(n+1)a_{n+2}+(3n(n-1)-2)a_n+3(n-1)a_{n-1}]x^n.
Exercise \PageIndex{13}
(1+2x^2)y''+(2-3x)y'+4y
- Answer
-
(1+2x^2)y''+(2-3x)y'+4y= \sum_{n=2}^\infty n(n-1)a_nx^{n-2} +2\sum_{n=2}^\infty n(n-1)a_nx^n +2\sum_{n=1}^\infty na_nx^{n-1}-2\sum_{n=0}^\infty a_nx^n +4\sum_{n=0}^\infty a_nx^n =\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n +2\sum_{n=0}^\infty n(n-1)a_nx^n +2\sum_{n=0}^\infty (n+1)a_{n+1}x^n-3\sum_{n=0}^\infty a_nx^n +4\sum_{n=0}^\infty a_nx^n= \sum_{n=0}^\infty \left[(n+2)(n+1)a_{n+2}+2(n+1)a_{n+1}+(2n^2-5n+4)a_n\right]x^n
Exercise \PageIndex{14}
(1+x^2)y''+(2-x)y'+3y
- Answer
-
(1+x^2)y''+(2-x)y'+3y= \sum_{n=2}^\infty n(n-1)a_nx^{n-2} +\sum_{n=2}^\infty n(n-1)a_nx^n +2\sum_{n=1}^\infty na_nx^{n-1}-\sum_{n=1}^\infty na_nx^n +3\sum_{n=0}^\infty a_nx^n =\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n \\+\sum_{n=0}^\infty n(n-1)a_nx^n +2\sum_{n=0}^\infty (n+1)a_{n+1}x^n-\sum_{n=0}^\infty na_nx^n +3\sum_{n=0}^\infty a_nx^n \\=\sum_{n=0}^\infty \left[(n+2)(n+1)a_{n+2}+2(n+1)a_{n+1}+(n^2-2n+3)a_n\right]x^n.
Exercise \PageIndex{15}
(1+3x^2)y''-2xy'+4y
Exercise \PageIndex{16}
Suppose y(x)=\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
\begin{eqnarray*} xy''+(4+2x)y'+(2+x)y. \end{eqnarray*}
- Answer
-
Let t=x+1; then xy''+(4+2x)y'+(2+x)y=(-1+t)y''+(2+2t)y'+(1+t)y =-\sum_{n=2}^\infty n(n-1)a_nt^{n-2} +\sum_{n=2}^\infty n(n-1)a_nt^{n-1} +2\sum_{n=1}^\infty n a_nt^{n-1} +2\sum_{n=1}^\infty n a_nt^n +\sum_{n=0}^\infty a_nt^n +\sum_{n=0}^\infty a_nt^{n+1} =-\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}t^n +\sum_{n=0}^\infty (n+1)na_{n+1}t^n +2\sum_{n=0}^\infty (n+1) a_{n+1}t^n +2\sum_{n=0}^\infty n a_nt^n +\sum_{n=0}^\infty a_nt^n +\sum_{n=1}^\infty a_{n-1}t^n =(-2a_2+2a_1+a_0) +\sum_{n=1}^\infty \left[-(n+2)(n+1)a_{n+2}+(n+1)(n+2)a_{n+1}+(2n+1)a_n+a_{n-1}\right] (x+1)^n.
Exercise \PageIndex{17}
Suppose y(x)=\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
\begin{eqnarray*} x^2y''+2xy'-3xy. \end{eqnarray*}
Exercise \PageIndex{18}
Do the following experiment for various choices of real numbers a_0 and a_1.
(a) Use differential equations software to solve the initial value problem
\begin{eqnarray*} (2-x)y''+2y=0,\quad y(0)=a_0,\quad y'(0)=a_1, \end{eqnarray*}
numerically on (-1.95,1.95). Choose the most accurate method your software package provides. (See Section 3.1 for a brief discussion of one such method.)
(b) For N=2, 3, 4, \dots, compute a_2, \dots, a_N from Equation (3.1.18) and graph
\begin{eqnarray*} T_N(x)=\sum_{n=0}^N a_nx^n \end{eqnarray*}
and the solution obtained in part (a) on the same axes. Continue increasing N until it's obvious that there's no point in continuing. (This sounds vague, but you'll know when to stop.)
Exercise \PageIndex{19}
Follow the directions of Exercise (3.1.18) for the initial value problem
\begin{eqnarray*} (1+x)y''+2(x-1)^2y'+3y=0,\quad y(1)=a_0,\quad y'(1)=a_1, \end{eqnarray*}
on the interval (0,2). Use Equations (3.1.24) \mbox{ and } (3.1.25) to compute \{a_n\}.
Exercise \PageIndex{20}
Suppose the series \sum_{n=0}^\infty a_nx^n converges on an open interval (-R,R), let r be an arbitrary real number, and define
\begin{eqnarray*} y(x)=x^r\sum_{n=0}^\infty a_nx^n=\sum_{n=0}^\infty a_nx^{n+r} \end{eqnarray*}
on (0,R). Use Theorem (3.1.4) and the rule for differentiating the product of two functions to show that
\begin{eqnarray*} y'(x)&=&\displaystyle{\sum_{n=0}^\infty (n+r)a_nx^{n+r-1}},\\ y''(x)&=&\displaystyle{\sum_{n=0}^\infty(n+r)(n+r-1)a_nx^{n+r-2}},\\ &\vdots&\\ y^{(k)}(x)&=&\displaystyle{\sum_{n=0}^\infty(n+r)(n+r-1)\cdots(n+r-k)a_nx^{n+r-k}} \end{eqnarray*}
on (0,R)
- Answer
-
y'(x)=\displaystyle x^r\sum_{n=0}^\infty na_nx^{n-1}+rx^{r-1}\sum_{n=0}^\infty a_nx^n=\sum_{n=0}^\infty (n+r)x^{n+r-1}
y''=\displaystyle{d\over dx}y'(x)={d\over dx}\left[x^{r-1}\sum_{n=0}^\infty (n+r)a_nx^n\right]=x^{r-1}\sum_{n=0}^\infty (n+r)na_nx^{n-1}+ (r-1)x^{r-2}\sum_{n=0}^\infty (n+r)a_nx^n=\sum_{n=0}^\infty (n+r)(n+r-1)a_nx^{n+r-2}.
In Exercises (3.1E.21) to (3.1E.26), let y be as defined in Exercise (3.1E.20), and write the given expression in the form x^r\sum_{n=0}^\infty b_nx^n.
Exercise \PageIndex{21}
x^2(1-x)y''+x(4+x)y'+(2-x)y
Exercise \PageIndex{22}
x^2(1+x)y''+x(1+2x)y'-(4+6x)y
- Answer
-
x^2(1+x)y''+x(1+2x)y'-(4+6x)y= (x^2y''+xy'-4y)+x(x^2y''+2xy'-6y)=\displaystyle \sum_{n=0}^\infty [(n+r)(n+r-1)+(n+r)-4]a_nx^{n+r} +\sum_{n=0}^\infty [(n+r)(n+r-1)+2(n+r)-6]a_nx^{n+r+1} =\sum_{n=0}^\infty (n+r-2)(n+r+2)a_nx^{n+r} +\sum_{n=0}^\infty (n+r+3)(n+r-2)a_nx^{n+r+1} =\sum_{n=0}^\infty (n+r-2)(n+r+2)a_nx^{n+r} +\sum_{n=1}^\infty (n+r+2)(n+r-3)a_{n-1}x^{n+r} =x^r\sum_{n=0}^\infty b_nx^n with
b_0=(r-2)(r+2)a_0 and
b_n=(n+r-2)(n+r+2)a_n+(n+r+2)(n+r-3)a_{n-1}, n\ge1.
Exercise \PageIndex{23}
x^2(1+x)y''-x(1-6x-x^2)y'+(1+6x+x^2)y
Exercise \PageIndex{24}
x^2(1+3x)y''+x(2+12x+x^2)y'+2x(3+x)y
- Answer
-
x^2(1+3x)y''+x(2+12x+x^2)y'+2x(3+x)y= (x^2y''+2xy')+x(3x^2y''+12xy'+6y)+x^2(xy'+2y)=\displaystyle \sum_{n=0}^\infty [(n+r)(n+r-1)+2(n+r)]a_nx^{n+r} +\sum_{n=0}^\infty [3(n+r)(n+r-1)+12(n+r)+6]a_nx^{n+r+1} +\sum_{n=0}^\infty[(n+r)+2]a_nx^{n+r+2 } =\sum_{n=0}^\infty (n+r)(n+r+1)a_nx^{n+r} +3\sum_{n=0}^\infty (n+r+1)(n+r+2)a_nx^{n+r+1} +\sum_{n=0}^\infty (n+r+2)a_nx^{n+r+2} =\sum_{n=0}^\infty (n+r)(n+r+1)a_nx^{n+r} +3\sum_{n=1}^\infty (n+r)(n+r+1)a_{n-1}x^{n+r} +\sum_{n=2}^\infty (n+r)a_{n-2}x^{n+r} =x^r\sum_{n=0}^\infty b_nx^n with
b_0=r(r+1)a_0, b_1=(r+1)(r+2)a_1+3(r+1)(r+2)a_0, b_n=(n+r)(n+r+1)a_n+3(n+r)(n+r+1)a_{n-1}+(n+r)a_{n-2}, n\ge2.
Exercise \PageIndex{25}
x^2(1+2x^2)y''+x(4+2x^2)y'+2(1-x^2)y
Exercise \PageIndex{26}
x^2(2+x^2)y''+2x(5+x^2)y'+2(3-x^2)y
- Answer
-
x^2(2+x^2)y''+2x(5+x^2)y'+2(3-x^2)y= (2x^2y''+10xy'+6y)+x^2(x^2y''+2xy'-2y)=\displaystyle \sum_{n=0}^\infty [2(n+r)(n+r-1)+10(n+r)+6]a_nx^{n+r} +\sum_{n=0}^\infty [(n+r)(n+r-1)+2(n+r)-2]a_nx^{n+r+2} =2\sum_{n=0}^\infty (n+r+1)(n+r+3)a_nx^{n+r} +\sum_{n=0}^\infty (n+r-1)(n+r+2)a_nx^{n+r+2} =2\sum_{n=0}^\infty (n+r+1)(n+r+3)a_nx^{n+r} +\sum_{n=2}^\infty (n+r-3)(n+r)a_{n-2}x^{n+r} =x^r\sum_{n=0}^\infty b_nx^n with
b_0=2(r+1)(r+3)a_0,
b_1=2(r+2)(r+4)a_1
b_n=2(n+r+1)(n+r+3)a_n+(n+r-3)(n+r)a_{n-2}, n\ .