University Calculus: Early Transcendentals (3rd Ed)
9.7
6.1.1 Definition
A power series is a function defined by
\(f(x)=\sum_{n=0}^\infty c_nx^n=c_0+c_1x+c_2x^2+c_3x^3+\dots\)
for a coefficient sequence \(\<c_n\>_{n=0}^\infty\).
Example
Expand the first four terms of the power series
\(\sum_{m=0}^\infty (2m+1)x^m\).
Example
Expand the first four terms of the power series
\(\sum_{k=2}^\infty \frac{x^{2k}}{k!}\).
The geometric series formula \(\sum_{n=0}^\infty ar^n\)
allows us to simplify certain series where \(|r|<1\).
Example
Simplify
\(f(x)=\sum_{n=0}^\infty x^n=1+x+x^2+x^3+\dots\) with domain \(|x|<1\).
Example
Simplify
\(p(x)=\sum_{j=1}^\infty (\frac{x}{3})^{2j}=
\frac{x^2}{9}+\frac{x^4}{81}+\frac{x^6}{729}+\dots\)
with domain \(|x|<3\).
6.1.2 Domains of Power Series
The domain of a power series may be determined by applying the Root or
Ratio Test to determine for which \(x\) values the series converges.
On the endpoints where these tests are inconclusive, other
techniques must be used.
Example
Find the domain of
\(f(x)=\sum_{n=1}^\infty\frac{x^n}{n}=x+\frac{x^2}{2}+\frac{x^3}{3}+\dots\).
Example
Find the domain of
\(h(x)=\sum_{n=0}^\infty\frac{(3-2x)^n}{n!}=
1+(3-2x)+\frac{(3-2x)^2}{2}+\frac{(3-2x)^3}{6}+\dots\).
Example
Find the domain of
\(g(x)=\sum_{k=2}^\infty\frac{(3x)^n}{n\ln n}=
\frac{9x^2}{2\ln2}+\frac{27x^3}{3\ln3}+\frac{81x^4}{4\ln4}+\dots\).
Exercises for 6.1
Expand the first four terms of the power series
\(\sum_{m=0}^\infty 3^{m+1}x^m\).
Expand the first four terms of the power series
\(\sum_{k=1}^\infty \frac{(-x)^k}{k+1}\).
Expand the first four terms of the power series
\(\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}\).
Simplify
\(q(x)=\sum_{n=1}^\infty (1-x)^n=(1-x)+(1-x)^2+(1-x)^3+\dots\)
with domain \(|1-x|<1\).
Simplify
\(g(x)=\sum_{j=0}^\infty (2x)^{2j+1}=
2x+8x^3+32x^5+128x^7+\dots\)
with domain \(|x|<\frac{1}{2}\).
Find the domain of
\(z(x)=\sum_{n=2}^\infty(-1)^n\frac{x^n}{2n}=
\frac{x^2}{4}-\frac{x^3}{6}+\frac{x^4}{8}-\frac{x^5}{10}+\dots\).
Find the domain of
\(f(x)=\sum_{i=0}^\infty\frac{(3x)^i}{(2i)!}=
1+\frac{3}{2}x+\frac{9}{24}x^2+\dots\).
Find the domain of
\(h(x)=\sum_{k=0}^\infty\frac{(x-2)^k}{k^2+1}=
1+\frac{x-2}{2}+\frac{(x-2)^2}{5}+\frac{(x-2)^3}{10}+\dots\).
(OPTIONAL)
Find the domain of
\(g(x)=\sum_{m=3}^\infty(\frac{1}{m}-\frac{1}{m+1})x^m\).
(QUIZ)
Expand the first four terms of the power series
\(\sum_{k=0}^\infty \frac{(-x)^{2k+1}}{(2k)!}\).
University Calculus: Early Transcendentals (3rd Ed)
9.8
6.2.1 Power Series from Functions
Let \(f(x)\) have derivatives of all orders nearby \(a\). Then the
Taylor series generated by \(f\) at \(a\) is given by
\(\sum_{k=0}^\infty\frac{f^{(k)}(a)}{k!}(x-a)^k\),
where \(f^{(k)}(a)\) is the \(k^{th}\) derivative of \(f\) at \(a\).
A Maclaurin series is a Taylor series where \(a=0\).
A Taylor/Maclaurin series is said to converge to its generating function if
it is equal to it for all members of its domain.
Example Let \(f(x)=\frac{1}{1+x}\) with the domain \(-1<x<1\),
and guess a formula for
\(f^{(k)}(0)\) by computing its first few terms. Then show that
the Maclaurin series generated by \(f\) converges to \(f\).
Example Let \(g(x)=\frac{2}{x}\) with the domain \(0<x<4\),
and guess a formula for
\(g^{(k)}(2)\) by computing its first few terms. Then show that
the Taylor series generated by \(g\) at \(2\) converges to \(g\).
It can be shown that \(f\) defined by \(f(0)=0\) and
\(f(x)=e^{-1/x^2}\) otherwise
satisfies \(f^{(k)}(0)=0\), giving an example
of a function which doesn’t converge to its Taylor series.
6.2.2 Maclaurin Series for \(e^x\), \(\sin x\), \(\cos x\)
The following Maclaurin Series can be shown to converge to their
generating functions:
\(\cos x = \sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}
= 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots\)
\(\sin x = \sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}
= x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+\dots\)
Example Show how to generate the Maclaurin series for \(e^x\).
Example Show how to generate the Maclaurin series for \(\sin x\).
6.2.3 Composition, Differentiation, and Integration of Power Series
Power series may be manipulated by the multiplication and composition
of continuous functions.
Example Find a power series converging to \(x\cos(-x^2)\).
Power series may be differentiated and integrated term-by-term.
Example Find a power series converging to \(\frac{1}{x^2-2x+1}\)
for \(-1<x<1\).
Example Find a power series converging to \(\tan^{\leftarrow}(x)\)
for \(-1<x<1\).
Exercises for 6.2
Let \(f(x)=\frac{1}{1-x}\) with the domain \(-1<x<1\),
and guess a formula for
\(f^{(k)}(0)\) by computing its first few terms. Then show that
the Maclaurin series generated by \(f\) converges to \(f\).
Let \(g(x)=\frac{3}{x}\) with the domain \(0<x<6\),
and guess a formula for
\(g^{(k)}(3)\) by computing its first few terms. Then show that
the Taylor series generated by \(g\) at \(3\) converges to \(g\).
Show how to generate the Maclaurin series
\(\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}\) for \(\cos x\).
Find the Maclaurin series for \(\sinh x\).
Find a power series converging to \(\frac{x^3}{e^{x^2}}\).
Find a power series converging to \(\frac{1}{x^2+2x+1}\)
for \(|x|<1\). (Hint: begin with the power series for
\(\frac{1}{1+x}\) and then differentiate term-by-term.)
Find a power series converging to \(\ln|x|\)
for \(0<x<2\). (Hint: begin with the power series for
\(\frac{1}{1+x}\) and then integrate term-by-term.)
(QUIZ) Generate the Maclaurin Series for \(\cosh x\).
University Calculus: Early Transcendentals (3rd Ed)
9.9
6.3.1 Taylor’s Formula
Taylor’s Formula guarantees that if \(f\) has derivatives of all orders
on an open interval containing \(a\), then for every nonnegative integer
\(n\) and \(x\) in that same interval,
\(f(x)=\left(\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k\right)+R_n(x)\)
where the error term is given by
\(R_n(x)=\frac{f^{(n+1)}(x_n)}{(n+1)!}(x-a)^{n+1}\) for some number
\(x_n\) between \(a\) and \(x\).
Example Use the fact that \(e<4\) and Taylor’s Formula to estimate
the value of \(\sqrt{e}\) with an error no greater than \(0.01\).
Example Use Taylor’s Formula to estimate
the value of \(\sin(0.1)\) with an error no greater than \(0.0001\).
6.3.2 Convergence of Taylor and Maclaurin Series
A Taylor series converges to its generating function when
\(\lim_{n\to\infty}|R_n(x)|=0\).
Example Prove that \(e^x=\sum_{k=0}^\infty\frac{x^k}{k!}\).
Example Prove that
\(\cos(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}\).
Exercises for 6.3
Use the fact that \(e<3\) and
Taylor’s Formula to estimate the value of \(e\) with an error
no greater than \(0.001\).
Use Taylor’s Formula to estimate the value of \(\cos(0.1)\) with an error
no greater than \(0.0001\).
Use Taylor’s Formula to estimate the value of \(\sin(1)\) with an error
no greater than \(0.01\).
Prove that \(\sin(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}\).
Use the fact that \(|\sinh(x_n)|\leq|\cosh(x_n)|\leq\cosh(x)\)
for any \(x_n\) between \(0\) and \(x\) to
prove that \(\cosh(x)=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}\).
Reprove \(\cosh(x)=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}\) by using
its definition \(\cosh(x)=\frac{1}{2}(e^x+e^{-x})\) along with
the Maclaurin series for \(e^x\).
(OPTIONAL) Prove that \(|\sinh(x_n)|\leq|\cosh(x_n)|\leq\cosh(x)\)
for any \(x_n\) between \(0\) and \(x\).
(QUIZ)
Which of these Maclaurin Series is most appropriate for approximating
\(e^{-1/2}=\frac{1}{\sqrt e}\)?
(QUIZ)
Find the error term \(R_n(x)\) from the Taylor Formula for \(e^x\),
where \(x_n\) is between \(0\) and \(x\).
\(R_n(x)=\frac{x^{n+1}}{n!}\)
\(R_n(x)=\frac{e^{x_n}}{(n+1)!}x^{n+1}\)
\(R_n(x)=\frac{1}{e^{x_n/2}(n+1)!}x^n\)
(QUIZ)
Use Taylor’s Formula to
approximate \(e^{-1/2}=\frac{1}{\sqrt e}\) with an error no greater than
\(\frac{1}{1000}=0.001\). (Hint: \(-1/2\leq x_n\leq 0\).)