4.1E: Exercises for Section 4.1

Exercise \(\PageIndex{1}\)

Is the power series \( \sum_{k=0}^\infty e^k x^k\) convergent? If so, what is the radius of convergence?

Exercise \(\PageIndex{2}\)

Is the power series \( \sum_{k=0}^\infty k x^k\) convergent? If so, what is the radius of convergence?

Exercise \(\PageIndex{3}\)

Is the power series \( \sum_{k=0}^\infty k! x^k\) convergent? If so, what is the radius of convergence?

Exercise \(\PageIndex{4}\)

Is the power series \( \sum_{k=0}^\infty \frac{1}{(2k)!} {(x-10)}^k\) convergent? If so, what is the radius of convergence?

Exercise \(\PageIndex{5}\)

Determine the Taylor series for \(\sin x\) around the point \(x_0 = \pi\).

Exercise \(\PageIndex{6}\)

Determine the Taylor series for \(\ln x\) around the point \(x_0 = 1\), and find the radius of convergence.

Exercise \(\PageIndex{7}\)

Determine the Taylor series and its radius of convergence of \(\dfrac{1}{1+x}\) around \(x_0 = 0\).

Exercise \(\PageIndex{8}\)

Determine the Taylor series and its radius of convergence of \(\dfrac{x}{4-x^2}\) around \(x_0 = 0\). Hint: You will not be able to use the ratio test.

Exercise \(\PageIndex{9}\)

Expand \(x^5+5x+1\) as a power series around \(x_0 = 5\).

Exercise \(\PageIndex{10}\)

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 \(\PageIndex{11}\)

Suppose that \(f\) is an analytic function such that \(f^{(n)}(0) = n\). Find \(f(1)\).

Exercise \(\PageIndex{12}\)

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 \(\PageIndex{13}\)

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 \(\PageIndex{14}\)

Using the geometric series, expand \(\frac{1}{1-x}\) around \(x_0=2\). For what \(x\) does the series converge?

Answer

\(\frac{1}{1-x}=-\frac{1}{1-(2-x)}\) so \(\frac{1}{1-x}=\sum\limits_{n=0}^\infty (-1)^{n+1}(x-2)^{n}\), which converges for \(1<x<3\).

Exercise \(\PageIndex{15}\)

Find the Taylor series for \(x^7 e^x\) around \(x_0 = 0\).

Answer

\(\sum\limits_{n=7}^\infty \frac{1}{(n-7)!} x^{n}\)

Exercise \(\PageIndex{16}\)

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.