Problem Sheet -- Expressions
620-295 Semester I 2010

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 28 February 2010

(1) Expressions
(2) Inverse expressions
(3) Expression identities
(4) Trig identities
(5) Inverse function identities
(6) Taylor series

Expressions

"What is" is a synonym of "Define".

	What is $x^2$?
	What is $e^x$?
	What is $\sin x$?
	What is $\cos x$?
	What is $\tan x$?
	What is $\cot x$?
	What is $\sec x$?
	What is $\csc x$?
	What is $\sinh x$?
	What is $\cosh x$?
	What is $\tanh x$?
	What is $\coth x$?
	What is $\mathrm{sech}x$?
	What is $\mathrm{csch}x$?

Inverse Expressions

	What is $\sqrt{x}$?
	What is $x^{\mathrm{1/2}}$?
	What is $\ln x$?
	What is $\log x$?
	What is ${\sin }^{-1}x$?
	What is $\arcsin x$?
	What is ${\cos }^{-1}x$?
	What is $\arccos x$?
	What is ${\tan }^{-1}x$?
	What is $\arctan x$?
	What is ${\cot }^{-1}x$?
	What is $\mathrm{arccot}x$?
	What is ${\sec }^{-1}x$?
	What is $\mathrm{arcsec}x$?
	What is ${\csc }^{-1}x$?
	What is $\mathrm{arccsc}x$?
	What is ${\sinh }^{-1}x$?
	What is $\mathrm{arcsinh}x$?
	What is ${\cosh }^{-1}x$?
	What is $\mathrm{arccosh}x$?
	What is ${\tanh }^{-1}x$?
	What is $\mathrm{arctanh}x$?
	What is ${\coth }^{-1}x$?
	What is $\mathrm{arccoth}x$?
	What is ${\mathrm{sech}}^{-1}x$?
	What is $\mathrm{arcsech}x$?
	What is ${\mathrm{csch}}^{-1}x$?
	What is $\mathrm{arccsch}x$?

Expression identities

"Explain why" is a synonym for "Prove that". "Verify the identity" is another synonym for "Prove that".

	Explain why ${\displaystyle \frac{1}{1-\mathrm{x}}=1+x+x^2+x^3+\cdots }$.
	Explain why ${\displaystyle \frac{x^n-1}{x-1}=1+x+x^2+\cdots +x^{n-1}}$.
	Find all possibilities for $c_0,c_1,c_2,\dots$ so that $f\left(x\right)=c_0+c_{1}x+c_2{x}^2+\cdots$ satisfies $f\left(x+y\right)=f\left(x\right)f\left(y\right)$.
	Explain why ${\displaystyle e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\cdots }$.
	Explain why $\ln x$ is the inverse function to $e^x$.
	Verify the identity $e^{x+y}=e^x{e}^y$.
	Verify the identity ${\displaystyle e^{-x}=\frac{1}{e^x}}$.
	Verify the identity ${\left(e^x\right)}^n=e^{nx}$.
	Verify the identity $e^0=1$.
	Verify the identity $\ln \left(xy\right)=\ln x+\ln y$.
	Verify the identity $-\ln x=\ln \left(1/\mathrm{x}\right)$.
	Verify the identity $\ln x^n=n\ln x$.
	Verify the identity $\ln 1=0$.
	Explain why ${\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots }$.
	Explain why ${\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots }$.
	Verify the identity $e^{ix}=\cos x+i\sin x$.
	Verify the identity ${\cos }^{2}x+{\sin }^{2}x=1$.
	Verify the identity $\sin \left(-x\right)=-\sin x$.
	Verify the identity $\cos \left(-x\right)=\cos x$.
	Verify the identity $\sin \left(x+y\right)=\sin x\cos y+\cos x\sin y$.
	Verify the identity $\cos \left(x+y\right)=\cos x\cos y-\sin x\sin y$.
	Verify the identity ${\displaystyle \cos x=\frac{e^{ix}+e^{-ix}}{2}}$.
	Verify the identity ${\displaystyle \sin x=\frac{e^{ix}-e^{-ix}}{2i}}$.
	Explain why ${\displaystyle \cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\cdots }$.
	Explain why ${\displaystyle \sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots }$.
	Verify the identity $e^x=\cosh x+\sinh x$.
	Verify the identity ${\cosh }^{2}x-{\sinh }^{2}x=1$.
	Verify the identity $\sinh \left(-x\right)=-\sinh x$.
	Verify the identity $\cosh \left(-x\right)=\cosh x$.
	Verify the identity $\sinh \left(x+y\right)=\sinh x\cosh y+\cosh x\sinh y$.
	Verify the identity $\cosh \left(x+y\right)=\cosh x\cosh y+\sinh x\sinh y$.
	Verify the identity ${\displaystyle \cosh x=\frac{e^x+e^{-x}}{2}}$.
	Verify the identity ${\displaystyle \sinh x=\frac{e^x+e^{-x}}{2}}$.
	Verify the identity ${\displaystyle \mathrm{arcsinh}x=\log (x+\sqrt{x^2+1})}$.
	Verify the identity ${\displaystyle \mathrm{arccosh}x=\log (x+\sqrt{x^2-1})}$.
	Verify the identity ${\displaystyle \mathrm{arctanh}t=\frac{1}{2}\log \left(\frac{1+t}{1-t}\right)}$.

Trig identities

	Verify the identity ${\displaystyle \tan \left(x+y\right)=\frac{\tan x+tany}{1-\tan x\tan y}}$.
	Verify the identity ${\displaystyle \sin \left(x/2\right)=\pm \sqrt{\frac{1-\cos x}{2}}}$.
	Verify the identity $\cos 3x={\cos }^{3}x-3\cos x{\sin }^{2}x$.
	Verify the identity $\sin 3x=3{\cos }^{2}x\sin x-{\sin }^{3}x$.
	Verify the identity ${\sin }^{2}A{\cot }^{2}A=\left(1-\sin A\right)\left(1+\sin A\right)$.
	Verify the identity ${\displaystyle \tan B=\frac{\cos B}{\sin B{\cot }^{2}B}}$.
	Verify the identity ${\displaystyle \frac{\tan V\cos V}{\sin V}=1}$.
	Verify the identity $\sin E\cot E+\cos E\tan E=\sin E+\cos E$.
	Verify the identity ${\displaystyle \frac{1}{{\sec }^{2}x}+\frac{1}{{\csc }^{2}x}-1=0}$.
	Verify the identity ${\displaystyle \frac{\sec A-1}{\sec A+1}+\frac{\cos A-1}{\cos A+1}=0}$.
	Verify the identity ${\displaystyle \sin V\left(1+{\cot }^{2}V\right)=\csc V}$.
	Verify the identity ${\displaystyle \frac{\sin \left(\pi /2-w\right)}{\cos \left(\pi /2-w\right)}=\cot w}$.
	Verify the identity ${\displaystyle \sec \left(\pi /2-z\right)=\frac{1}{\sin z}}$.
	Verify the identity ${\displaystyle 1+{\tan }^2\left(\pi /2-x\right)=\frac{1}{{\cos }^2\left(\pi /2-x\right)}}$.
	Verify the identity ${\displaystyle \frac{\sin A}{\csc A}+\frac{\cos A}{\sec A}=1}$.
	Verify the identity ${\displaystyle \frac{\sec B}{\cos B}-\frac{\tan B}{\cot B}=0}$.
	Verify the identity ${\displaystyle \frac{1}{{\csc }^{2}w}+{\sec }^{2}w+\frac{1}{{\sec }^{2}w}=2+\frac{{\sec }^{2}w}{{\csc }^{2}w}}$.
	Verify the identity ${\displaystyle {\sec }^{4}V-{\sec }^{2}V=\frac{1}{{\cot }^{4}V}+\frac{1}{{\cot }^{2}V}}$.
	Verify the identity ${\displaystyle {\sin }^{4}x+{\cos }^{2}x={\cos }^{4}x+{\sin }^{2}x}$.
	Verify the identity ${\displaystyle \tan 3\alpha =\frac{3\tan \alpha -{\tan }^3\alpha }{1-3{\tan }^2\alpha }}$.
	Verify the identity ${\displaystyle \cot \left(\alpha /2\right)=\frac{\sin \alpha }{1-\cos \alpha }}$.
	Verify the identity ${\displaystyle \cos \left(\pi /6-x\right)+\cos \left(\pi /6+x\right)=\sqrt{3}\cos x}$.
	Verify the identity ${\displaystyle \sin \left(\alpha +\beta \right)\sin \left(\alpha -\beta \right)={\sin }^2\alpha -{\sin }^2\beta }$.
	Verify the identity ${\displaystyle \sin \left(\pi /3-x\right)+\sin \left(\pi /3+x\right)=\sqrt{3}\cos x}$.
	Verify the identity ${\displaystyle \cos \left(\pi /4-x\right)-\cos \left(\pi /4+x\right)=\sqrt{2}\sin x}$.
	Verify the identity ${\displaystyle 2\sin \alpha \cos \beta =\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)}$.
	Verify the identity ${\displaystyle 2\sin \alpha \sin \beta =\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)}$.
	Verify the identity ${\displaystyle \cos 2\theta =2\sin \left(\pi /4+\theta \right)\sin \left(\pi /4-\theta \right)}$
	Verify the identity ${\displaystyle \frac{\sin 2A}{2}=\frac{\tan A}{1+{\tan }^{2}A}}$.
	Verify the identity ${\displaystyle \cot \left(x/2\right)=\frac{1+\cos x}{\sin x}}$.
	Verify the identity ${\displaystyle \sin 2B\left(\cot B+\tan B\right)=2}$.
	Verify the identity ${\displaystyle \frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }=\cos 2\theta }$.
	Verify the identity ${\displaystyle 1+\cos 2A=\frac{2}{1+{\tan }^{2}A}}$.
	Verify the identity ${\displaystyle \tan 2x\tan x+2=\frac{\tan 2x}{\tan x}}$.
	Verify the identity ${\displaystyle \csc A\sec A=2\csc 2A}$.
	Verify the identity ${\displaystyle \cot x=\frac{\sin 2x}{1-\cos 2x}}$.
	Verify the identity ${\displaystyle 1-\sin A={\left(\sin \frac{A}{2}-\cos \frac{A}{2}\right)}^2}$.
	Verify the identity ${\displaystyle {\cos }^{4}A=\frac{2\cos 2A+{\cos }^{2}2A\mathrm{+}1}{4}}$.
	Verify the identity ${\displaystyle \frac{\sin A+\sin B}{\sin A-\cos A}=\frac{\tan \left(\frac{A+B}{2}\right)}{\tan \left(\frac{A-B}{2}\right)}}$.
	Verify the identity ${\displaystyle \frac{\sin \alpha +\sin 3\alpha }{\cos \alpha +\cos 3\alpha }=\tan 2\alpha }$.
	Verify the identity ${\displaystyle \frac{\cos 2A}{1+\sin 2A}=\frac{\cot A-1}{\cot A+1}}$.
	Verify the identity ${\displaystyle \frac{\cos A+\sin A}{\cos A-\sin A}=\frac{1+\sin 2A}{\cos 2A}}$.
	Verify the identity ${\displaystyle \cot \alpha -\cot \beta =\frac{\sin \left(\beta -\alpha \right)}{\sin \alpha \sin \beta }}$.
	Verify the identity ${\displaystyle \tan \theta \csc \theta \cos \theta =1}$.
	Verify the identity ${\displaystyle {\cos }^2\theta =\frac{{\cot }^2\theta }{1+{\cot }^2\theta }}$.
	Verify the identity ${\displaystyle \frac{1-\sin A}{1+\sin A}={\left(\sec A-\tan A\right)}^2}$.
	Verify the identity ${\displaystyle {\left(\tan A-\cot A\right)}^2+4={\sec }^{2}A+{\csc }^{2}A}$.
	Verify the identity ${\displaystyle \cos B\cos \left(A+B\right)+\sin B\sin \left(A+B\right)=\cos B}$.
	Verify the identity ${\displaystyle \frac{\tan A-\sin A}{\sec A}=\frac{{\sin }^{3}A}{1+\cos A}}$.
	Verify the identity ${\displaystyle \frac{2{\tan }^{2}A}{1+{\tan }^{2}A}=1-\cos 2A}$.
	Verify the identity ${\displaystyle \tan 2A=\tan A+\frac{\tan A}{\cos 2A}}$.
	Verify the identity ${\displaystyle \sin 2A=\frac{2\tan A}{1+{\tan }^{2}A}}$.
	Verify the identity ${\displaystyle \frac{4\sin A}{1-{\sin }^{2}A}=\frac{1+\sin A}{1-\sin A}-\frac{1-\sin A}{1+\sin A}}$.
	Verify the identity ${\displaystyle \tan A+\sin A=\frac{\csc A+\cot A}{\csc A\cot A}}$.

Inverse function identities

	Verify the identity ${\displaystyle \cos \left({\tan }^{-1}x\right)=\frac{1}{\sqrt{1+x^2}}}$.
	Verify the identity ${\displaystyle \sin \left({\tan }^{-1}x\right)=\frac{x}{\sqrt{1+x^2}}}$.
	Verify the identity ${\displaystyle \sin \left({\cos }^{-1}x\right)=\sqrt{1-x^2}}$.
	Verify the identity ${\displaystyle \tan \left({\cos }^{-1}x\right)=\frac{\sqrt{1-x^2}}{x}}$.
	Verify the identity ${\displaystyle \cos \left({\sin }^{-1}x\right)=\sqrt{1-x^2}}$.
	Verify the identity ${\displaystyle \tan \left({\cot }^{-1}x\right)=\frac{1}{x}}$.
	Verify the identity ${\displaystyle \cot \left({\cot }^{-1}2\right)=2}$.
	Verify the identity ${\displaystyle \sin \left({\cot }^{-1}x\right)=\frac{1}{\sqrt{1+x^2}}}$.
	Verify the identity ${\displaystyle \cos \left({\cot }^{-1}x\right)=\frac{x}{\sqrt{1+x^2}}}$.
	Verify the identity ${\displaystyle {\sin }^{-1}\left(-x\right)=-{\sin }^{-1}x}$.
	Verify the identity ${\displaystyle {\tan }^{-1}\left(-x\right)=-{\tan }^{-1}x}$.
	Verify the identity ${\displaystyle {\tan }^{-1}x={\cot }^{-1}\left(1/x\right)}$.
	Verify the identity ${\displaystyle {\tan }^{-1}x={\sin }^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)}$.
	Verify the identity ${\displaystyle {\sin }^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)={\cos }^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)}$.
	Verify the identity ${\displaystyle \mathrm{arcsinh}\left(\frac{x}{\sqrt{1-x^2}}\right)=\mathrm{arctanh}x}$.

Taylor series

	Define the set $\mathbb{Q}\left[x\right]$ and give three example of elements of $\mathbb{Q}\left[x\right]$.
	Define the set $\mathbb{Q}[[x]]$ and give three example of elements of $\mathbb{Q}\left[\right[x\left]\right]$.
	Define the set $\mathbb{Q}\left(x\right)$ and give three examples of elements of $\mathbb{Q}\left(x\right)$.
	Define the set $\mathbb{Q}((x))$ and give three examples of elements of $\mathbb{Q}\left(\right(x\left)\right)$.
	Let $D:\mathbb{Q}\left[x\right]\to \mathbb{Q}\left[x\right]$ be a function such that (D1) If $f,g\in \mathbb{Q}\left[x\right]$ then $D(f+g)=D\left(f\right)+D\left(g\right)$, (D2) If $c\in \mathbb{Q}$ and $f\in \mathbb{Q}\left[x\right]$ then $D\left(cf\right)=cD\left(f\right)$, (D3) If $f,g\in \mathbb{Q}\left[x\right]$ then $D\left(fg\right)=fD\left(g\right)+D\left(f\right)g$ and (D4) $D\left(x\right)=1$. Prove that if $n\in {\mathbb{Z}}_{>0}$ then $D\left(x^n\right)=n{x}^{n-1}$.
	Write out the first ten terms of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n+1}}$.
	Write out the first ten terms of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n\text{!}}}$.
	Write out the first ten terms of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{(x-1)}^n}{n}}$.
	Write out the first ten terms of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{x^n}{n+2}}$.
	Let $D:\mathbb{Q}\left[\right[x\left]\right]\to \mathbb{Q}\left[\right[x\left]\right]$ be a function such that (D1) If $f,g\in \mathbb{Q}\left[\right[x\left]\right]$ then $D(f+g)=D\left(f\right)+D\left(g\right)$, (D2) If $c\in \mathbb{Q}$ and $f\in \mathbb{Q}\left[\right[x\left]\right]$ then $D\left(cf\right)=cD\left(f\right)$, (D3) If $f,g\in \mathbb{Q}\left[\right[x\left]\right]$ then $D\left(fg\right)=fD\left(g\right)+D\left(f\right)g$ and (D4) $D\left(x\right)=1$. Prove that if $f=c_0+c_{1}x+c_2{x}^2+c_3{x}^3+c_4{x}^4+⃛$ then $c_k=\frac{1}{k!}\left(D^{k}f\right){|}_{x=0}$.
	Write $\frac{1-x^n}{1-x}$ as an element of $\mathbb{Q}\left[x\right]$.
	Write $e^x$ as an element of $\mathbb{Q}\left[\right[x\left]\right]$.
	Write $\sin x$ as an element of $\mathbb{Q}\left[\right[x\left]\right]$.
	Write $\sin (1+x)$ as an element of $\mathbb{Q}\left[\right[x\left]\right]$.
	Write $\cos x$ as an element of $\mathbb{Q}\left[\right[x\left]\right]$.
	Write $\frac{1}{1-x}$ as an element of $\mathbb{Q}\left[\right[x\left]\right]$.
	Write ${(1+x)}^7$ as an element of $\mathbb{Q}\left[\right[x\left]\right]$.
	Write ${(1+x)}^{\mathrm{1/7}}$ as an element of $\mathbb{Q}\left[\right[x\left]\right]$.
	Find the power series representation of ${\displaystyle \frac{1}{1+2x}}$.
	Find a series representation of ${\displaystyle \frac{1}{1+x^2}}$.
	Find a series representation of ${\displaystyle \frac{x}{1+x}}$.
	Find the series representation of ${\displaystyle \frac{1}{{(1+x)}^2}}$.
	Find a series representation of $\arctan x$.
	Find a series representation of $\log (2+x)$.
	Find a series expansion of $\cosh x$.
	Find a series expansion of ${\displaystyle \frac{1}{1-x}}$.
	Find a series expansion of $\log (1-x)$.
	Find a series expansion of $\log (1+x)$.
	Find a series expansion of ${\displaystyle \frac{\log (1+x)}{\log (1-x)}}$.
	Find a series expansion of $\sinh x$.
	Find a series expansion of $\log (1+2x)$.
	Find a series expansion of $\sin x$.
	Find a series expansion of $\sin \left(2x\right)$.
	Find a series expansion of $\cos x$.
	Find a series expansion of ${\displaystyle \frac{1}{1+x}}$.
	Find a series expansion of $\sinh x$.
	Find a series expansion of $\log (2x+1)$.
	Find a series expansion of ${(1+x)}^{-2}$.
	Find a series expansion of $\sin \left({\theta }^2\right)$.
	Find a series expansion of $x\sin 3x$.
	Find a series expansion of ${\cos }^{2}x$.
	Find a series expansion of ${\displaystyle \frac{t}{1+t}}$.
	Find a series expansion of ${\displaystyle \frac{z}{e^{2z}}}$.
	Find a series expansion of $\sin \left(x^2\right)$.
	Find a series representation of ${\displaystyle \int e^{x^3}dx}$.
	Find the power series representation of ${\displaystyle \int \frac{\sinh x}{x}dx}$.
	Find an infinite series representation of ${\displaystyle {\int }_{-1}^1\frac{\sinh x}{x}dx}$.
	Find a series expansion of ${\displaystyle \int \frac{\cosh x-1}{x^2}.}\hspace{0.17em}dx$.
	Find an infinite series representation of ${\displaystyle {\int }_0^1{e}^{x^3}dx}$.
	Find a series expansion of ${\displaystyle {\int }_0^{\mathrm{t}}\sin \left(x^2\right)\hspace{0.17em}dx}$
	Find the Taylor expansion of $e^x$ at $x=0$.
	Find the Taylor expansion of $\sinh x$ at $x=0$.
	Find the Taylor expansion of ${\displaystyle \frac{1}{1-x}}$ at $x=0$.
	Find the Taylor expansion of $e^x$ at $x=2$.
	Find the Taylor expansion of $\log x$ at $x=1$.
	Find the Taylor expansion of ${\displaystyle \frac{1}{x^2}}$ at $x=1$.
	Find the Taylor series for $\sin x$ at the point $a=\frac{1}{4}\pi$.
	Find the Taylor series for $\cos x$ at the point $a=\frac{1}{3}\pi$.
	Find the Taylor series for $\frac{1}{x}$ at the point $a=2$.
	Find the Taylor series for $e^x$ at the point $a=-3$.
	Find a series representation for $e^{2x}$ in powers of $x+1$.
	Find a series representation for $\log x$ in powers of $x-1$.
	Find an alternate expression for the series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}n{x}^n}$.
	Find an alternate expression for the series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{n}{2^n}}$.
	Find an alternate expression for the series ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}n(n-1)x^n}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }n{x}^{n-1}}$.
	Find the sum of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^{n+1}}{n+1}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{x^n}{n}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n}{3^{n-1}}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n{2}^{n+1}}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }n(n-1){\left(\frac{1}{4}\right)}^n}$.
	Find an alternate expression for the series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}n{x}^{n-1}}$.
	Find an alternate expression for the series ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}n(n-1)x^{n-2}}$.
	Find an alternate expression for the series ${\displaystyle \sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{2^k{x}^k}{k}}$.

References

[Ca] S. Carnie, 620-143 Applied Mathematics, Course materials, 2006 and 2007.

[Ho] C. Hodgson, 620-194 Mathematics B and 620-211 Mathematics 2 Notes, Semester 1, 2005.

[Wi] P. Wightwick, UMEP notes, 2010.