Chapter 13

a) Define the general form, radius of convergence, and interval of convergence of a power series. b) Given the power series \(\sum a_{n} x^{n}\) show that the series converges, if given that: i) \(\mathrm{R}=1 / \mathrm{L}\) where \(\mathrm{L}=\lim _{\mathrm{n} \rightarrow \infty}\left[\left(\mathrm{a}_{\mathrm{n}+1}\right) /\left(\mathrm{a}_{\mathrm{n}}\right)\right]\) exists and \(|\mathrm{x}|<\mathrm{R}\) or ii) \(\mathrm{R}=1 / \alpha\) where \(\alpha=\lim _{\mathrm{n} \rightarrow \infty}\) sup \(\left(\left.\mathrm{n} \sqrt{\mid}\right|_{n} \mid\right)\) and \(|\mathrm{x}|<\mathrm{R}\).

Find the radius of convergence of the following power series: a) \(^{\infty} \sum_{n=1}\left(x^{n} / n\right)\) b) \({ }^{\infty} \Sigma_{\mathrm{n}=1}\left\\{(2 \mathrm{n}) ! /(\mathrm{n} !)^{2}\right\\} \mathrm{x}^{\mathrm{n}}\) c) \(^{\infty} \Sigma_{\mathrm{n}=1}\left\\{(3 \mathrm{n}) ! /(\mathrm{n} !)^{2}\right\\} \mathrm{x}^{\mathrm{n}}\)

Considering all possibilities, find the radius of convergence of the series $$ \sum\left[(\mathrm{pn}) ! /(\mathrm{n} !)^{9}\right] \mathrm{x}^{\mathrm{n}} $$ where \(p\) is a positive integer and \(q>0\).

Find the radius of convergence of the power series $$ \infty \sum_{n=1}\left(x^{n} / n^{2}\right) $$ Then determine if the convergence is uniform for \(-\mathrm{R} \leq \mathrm{x} \leq \mathrm{R}\)

Prove Abel's theorem which states: If the series $$ { }^{\infty} \sum_{n=0} a_{n} x^{n} $$ converges at \(R\), then it converges uniformly on the closed interval \(0 \leq \mathrm{x} \leq \mathrm{R}\). (A like conclusion holds for \(-\mathrm{R} \leq \mathrm{x} \leq 0\) if the series converges at \(\mathrm{x}=-\mathrm{R}\) ).

Prove Abel's limit theorem which states: If $$ { }^{\infty} \sum_{n=0} a_{n} x^{n} $$ converges at \(\mathrm{x}=\mathrm{x}_{0}\), where \(\mathrm{x}_{0}\) may be an interior point or an endpoint of the interval of convergence, then \(\lim _{(x \rightarrow(x)) 0}\left\\{^{\infty} \sum_{n=0} a_{n} x^{n}\right\\}={ }^{\infty} \sum_{n=0} \lim _{(x \rightarrow(x)) 0} a_{n} x^{n}={ }^{\infty} \sum_{n=0} a_{n} x_{0}{\underline{\phantom{xx}}}^{n}\) If \(\mathrm{x}_{0}\) is a left hand endpoint it is proper to use \(\mathrm{x} \rightarrow \mathrm{x}_{0}^{+}\) and for a right hand endpoint \(\mathrm{x} \rightarrow \mathrm{x}_{0}\)

a) Using power series, show that \([\mathrm{d}(\sin \mathrm{x}) / \mathrm{d} \mathrm{x}]=\cos \mathrm{x}\) and \([\mathrm{d}(\cos \mathrm{x}) / \mathrm{d} \mathrm{x}]=-\sin \mathrm{x}\) b) Then show that \(\sin \mathrm{a} \cos \mathrm{b}+\cos \mathrm{a} \sin \mathrm{b}=\sin (\mathrm{a}+\mathrm{b})\) and \(\cos a \cos b-\sin a \sin b=\cos (a+b)\)

Show that the series representation $$ \mathrm{e}^{\mathrm{x}}=1+\mathrm{x}+\left(\mathrm{x}^{2} / 2 !\right)+\left(\mathrm{x}^{3} / 3 !\right)+\left(\mathrm{x}^{\mathrm{n}} / \mathrm{n} !\right)+\ldots $$ is valid for all values of \(\mathrm{x}\).

Derive the series expansion $$ \begin{aligned} \sin ^{-1} x=x+(1 / 2)\left(x^{3} / 3\right)+(1 / 2) \cdot(3 / 4)\left(x^{5} / 5\right) \\ &+[(1 \cdot 3 \cdot 5) /(2 \cdot 4 \cdot 6)]\left(x^{7} / 7\right)+\ldots . \end{aligned} $$

a) Find an expansion in powers of \(\mathrm{x}\) of the function $$ \mathrm{f}(\mathrm{x})={ }^{1} \int_{0}\left[\left(1-\mathrm{e}^{-\mathrm{tx}}\right) / \mathrm{t}\right] \mathrm{dt} \text { . } $$ b) Use the results from part (a) to find \(\mathrm{f}(1 / 2)\) approximately.

Find a power series in \(\mathrm{x}\) for: a) \(\tan \mathrm{x}\) b) \([(\sin x) /(\sin 2 x)]\) \((\mathrm{x} \neq 0)\)

Show that \(1 \int_{0}[\\{\log (1-t)\\} / t] \mathrm{dt}=-\left[\left(1 / 1^{2}\right)+\left(1 / 2^{2}\right)+\left(1 / 3^{2}\right)+\ldots\right.\) \(\left.+\left(1 / n^{2}\right)+\ldots\right]\)

Let the functions \(\mathrm{J}_{0}(\mathrm{x}), \mathrm{J}_{1}(\mathrm{x})\) be defined as follows: \(\mathrm{J}_{0}(\mathrm{x})=1-\left[\mathrm{x}^{2} /\left\\{(1 !)^{2} 2^{2}\right\\}\right]+\left[\mathrm{x} /\left\\{(2 !)^{2} 2^{4}\right\\}\right]-\ldots\) \(+(-1)^{\mathrm{n}}\left[\mathrm{x}^{2 \mathrm{n}} /\left\\{(\mathrm{n} !)^{2} 2^{2 \mathrm{n}}\right\\}\right]+\ldots\) \(\mathrm{J}_{1}(\mathrm{x})=(\mathrm{x} / 2)\left[1-\left\\{\left(\mathrm{x}^{2}\right) /(1 ! 2 ! 3 !)^{2}\right\\}+\ldots+\left\\{(-1)^{\mathrm{n}}\right\\}\right.\) \(\left.\left\\{\left(\mathrm{x}^{2 \mathrm{n}}\right) /\left\\{\mathrm{n} !(\mathrm{n}+1) ! 2^{\mathrm{n}}\right\\}\right\\}\right]\) Show that \(\mathrm{J}_{0}(\mathrm{x}), \mathrm{J}_{1}(\mathrm{x})\) are defined for all values of \(\mathrm{x}\) and that \(\mathrm{J}_{0}^{\prime}(\mathrm{x})=-\mathrm{J}_{1}(\mathrm{x})\)

Using Power series, show that $$ \log 2=1-(1 / 2)+(1 / 3)-(1 / 4)+\ldots $$

By the use of power series, show that for \(\mathrm{x}, \mathrm{y} \in \mathrm{R}\) : a) \(e^{x} e^{y}=e^{x+y}\) b) \(\sin x \cos x=(1 / 2) \sin 2 x\)