Exercise \(\PageIndex{7.1.1}\)
Is the power series \( \sum_{k=0}^\infty e^k x^k\) convergent? If so, what is the radius of convergence?
Exercise \(\PageIndex{7.1.2}\)
Is the power series \( \sum_{k=0}^\infty k x^k\) convergent? If so, what is the radius of convergence?
Exercise \(\PageIndex{7.1.3}\)
Is the power series \( \sum_{k=0}^\infty k! x^k\) convergent? If so, what is the radius of convergence?
Exercise \(\PageIndex{7.1.4}\)
Is the power series \( \sum_{k=0}^\infty \frac{1}{(2k)!} {(x10)}^k\) convergent? If so, what is the radius of convergence?
Exercise \(\PageIndex{7.1.5}\)
Determine the Taylor series for \(\sin x\) around the point \(x_0 = \pi\).
Exercise \(\PageIndex{7.1.6}\)
Determine the Taylor series for \(\ln x\) around the point \(x_0 = 1\), and find the radius of convergence.
Exercise \(\PageIndex{7.1.7}\)
Determine the Taylor series and its radius of convergence of \(\dfrac{1}{1+x}\) around \(x_0 = 0\).
Exercise \(\PageIndex{7.1.8}\)
Determine the Taylor series and its radius of convergence of \(\dfrac{x}{4x^2}\) around \(x_0 = 0\). Hint: You will not be able to use the ratio test.
Exercise \(\PageIndex{7.1.9}\)
Expand \(x^5+5x+1\) as a power series around \(x_0 = 5\).
Exercise \(\PageIndex{7.1.10}\)
Suppose that the ratio test applies to a series \( \sum_{k=0}^\infty a_k x^k\). Show, using the ratio test, that the radius of convergence of the differentiated series is the same as that of the original series.
Exercise \(\PageIndex{7.1.11}\)
Suppose that \(f\) is an analytic function such that \(f^{(n)}(0) = n\). Find \(f(1)\).
Exercise \(\PageIndex{7.1.12}\)
Is the power series \( \sum_{n=1}^\infty {(0.1)}^n x^n\) convergent? If so, what is the radius of convergence?
 Answer

Yes. Radius of convergence is \(10\).
Exercise \(\PageIndex{7.1.13}\): (challenging)
Is the power series \( \sum_{n=1}^\infty \frac{n!}{n^n} x^n\) convergent? If so, what is the radius of convergence?
 Answer

Yes. Radius of convergence is \(e\).
Exercise \(\PageIndex{7.1.14}\)
Using the geometric series, expand \(\frac{1}{1x}\) around \(x_0=2\). For what \(x\) does the series converge?
 Answer

\(\frac{1}{1x}=\frac{1}{1(2x)}\) so \(\frac{1}{1x}=\sum\limits_{n=0}^\infty (1)^{n+1}(x2)^{n}\), which converges for \(1<x<3\).
Exercise \(\PageIndex{7.1.15}\): (challenging)
Find the Taylor series for \(x^7 e^x\) around \(x_0 = 0\).
 Answer

\(\sum\limits_{n=7}^\infty \frac{1}{(n7)!} x^{n}\)
Exercise \(\PageIndex{7.1.16}\): (challenging)
Imagine \(f\) and \(g\) are analytic functions such that \(f^{(k)}(0) = g^{(k)}(0)\) for all large enough \(k\). What can you say about \(f(x)g(x)\)?
 Answer

\(f(x)g(x)\) is a polynomial. Hint: Use Taylor series.
Exercise \(\PageIndex{7.3.1}\)
Find a particular (Frobeniustype) solution of \(x^2 y'' + x y' + (1+x) y = 0\).
Exercise \(\PageIndex{7.3.2}\)
Find a particular (Frobeniustype) solution of \(x y''  y = 0\).
Exercise \(\PageIndex{7.3.3}\)
Find a particular (Frobeniustype) solution of \(y'' +\frac{1}{x}y'  xy = 0\).
Exercise \(\PageIndex{7.3.4}\)
Find the general solution of \(2 x y'' + y'  x^2 y = 0\).
Exercise \(\PageIndex{7.3.5}\)
Find the general solution of \(x^2 y''  x y' y = 0\).
Exercise \(\PageIndex{7.3.6}\)
In the following equations classify the point \(x=0\) as ordinary, regular singular, or singular but not regular singular.
 \(x^2(1+x^2)y''+xy=0\)
 \(x^2y''+y'+y=0\)
 \(xy''+x^3y'+y=0\)
 \(xy''+xy'e^xy=0\)
 \(x^2y''+x^2y'+x^2y=0\)
Exercise \(\PageIndex{7.3.7}\)
In the following equations classify the point \(x=0\) as ordinary, regular singular, or singular but not regular singular.
 \(y''+y=0\)
 \(x^3y''+(1+x)y=0\)
 \(xy''+x^5y'+y=0\)
 \(\sin(x)y''y=0\)
 \(\cos(x)y''\sin(x)y=0\)
 Answer

 ordinary,
 singular but not regular singular,
 regular singular,
 regular singular,
 ordinary.
Exercise \(\PageIndex{7.3.8}\)
Find the general solution of \(x^2 y'' y = 0\).
 Answer

\(y=Ax^{\frac{1+\sqrt{5}}{2}}+Bx^{\frac{1\sqrt{5}}{2}}\)
Exercise \(\PageIndex{7.3.9}\)
Find a particular solution of \(x^2 y'' +(x\frac{3}{4})y = 0\).
 Answer

\(y=x^{3/2}\sum\limits_{k=0}^\infty \frac{(1)^{1}}{k!(k+2)!}x^{k}\) (Note that for convenience we did not pick \(a_{0}=1\).)
Exercise \(\PageIndex{7.3.10}\): (tricky)
Find the general solution of \(x^2 y''  x y' +y = 0\).
 Answer

\(y=Ax+Bx\ln (x)\)