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The series representation of a function \(f\) given by the power series \(f(x) =f(c)+f^{\prime} (c)(x-c)+\frac{f^{\prime \prime} (c) (x-c) ^{2}}{2!}+\cdots+\) \(\frac{f^{(n) }(c)\,(x-c)^{n}}{n!}+\cdots\) is called a(n)______________ ______________ about \(c\).

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If \(c=0\) in the Taylor expansion of a function \(f,\) then the expansion is called a(n) ______________ expansion.

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\(f(x)=\ln (1-x)\)

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\(f(x)=\ln (1+x)\)

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\(f(x)=\frac{1}{1-x}\)

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\(f(x)=\frac{1}{1-3x}\)

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\(f(x)=\frac{1}{(1+x)^{2}}\)

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\(f(x)=(1+x) ^{-3}\)

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\(f(x)=\frac{1}{1+x^{2}}\)

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\(f(x)=\frac{1}{1+2x^{3}}\)

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\(f(x)=e^{3x}\)

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\(f(x)=e^{x/2}\)

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\(f(x)=\sin (\pi x)\)

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\(f(x)=\cos (2x)\)

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\(f(x)=e^{x};\quad c=1\)

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\(f(x)=e^{2x};\quad c=-1\)

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\(f(x)=\ln x;\quad c=1\)

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\(f(x)=\sqrt{x};\quad c=1\)

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\(f(x)=\frac{1}{x};\quad c=1\)

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\(f(x)=\frac{1}{\sqrt{x}};\quad c=4\)

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\(f(x)=\sin x;\quad c=\frac{\pi }{6}\)

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\(f(x)=\cos x;\quad c=-\frac{\pi }{2}\)

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\(f(x)=3x^{3}+2x^{2}+5x-6;\quad c=0\)

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\(f(x)=4x^{4}-2x^{3}-x;\quad c=0\)

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\(f(x)=3x^{3}+2x^{2}+5x-6;\quad c=1\)

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\(f(x)=4x^{4}-2x^{3}+x;\quad c=1\)

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\(f(x)=\sinh x\)

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\(f(x)= e^{-x^{2}}\)

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\(f(x)=xe^{x}\)

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\(f(x)=xe^{-x}\)

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\(f(x)=e^{-x}\sin x\)

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\(f(x)=e^{-x}\cos x\)

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\(f(x)=\frac{1}{\sqrt{1-x^{2}}};\quad g(x)=\sin ^{-1}x\)

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\(f(x)=\tan x;\quad g(x)=\ln (\cos x)\)

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\(f(x)= \sqrt{1+x^{2}}\)

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\(f(x)=\frac{1}{\sqrt{1-x}}\)

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\(f(x)=(1+x)^{1/5}\)

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\(f(x)=(1-x) ^{5/3}\)

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\(f(x)=\frac{1}{(1+x^{2}) ^{1/2}}\)

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\(f(x)=\frac{1}{(1+x)^{3/4}}\)

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\(f(x)= \frac{2x}{\sqrt{1-x}}\)

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\(f(x)=\frac{x}{1+x ^{3}}\)

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Find the Maclaurin expansion for \(f(x) =\sin ^{2}x.\)

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Find the Maclaurin expansion for \(f(x) =\cos ^{2}x.\)

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Obtain the Maclaurin expansion of \(\cos x\) by integrating the Maclaurin series for \(\sin x\).

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Find the Maclaurin expansion for \(f(x) =\ln \frac{1}{1-x}\). Compare the result to the power series representation of \(f(x) =\ln \frac{1}{1-x}\) found in Section 8.8, Example 8, page 607.

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Find the first five nonzero terms of the Maclaurin expansion for \(f(x) =\sec x.\)

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Probability The standard normal distribution \(p(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2\!/2}\) is important in probability and statistics. If a random variable \(Z\) has a standard normal distribution, then the probability that an observation of \(Z\) is between \(Z=a\) and \(Z=b\) is given by \[ P(a\leq Z\leq b) =\frac{1}{\sqrt{2\pi}}\, \int_{a}^{b} e^{-x^2\!/2}\, dx \]

1. Find the Maclaurin expansion for \(p(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2\!/2}\).
2. Use properties of power series to find a power series representation for \(P\).
3. Use the first four terms of the series representation for \(P\) to approximate \(P(-0.5\leq Z\leq 0.3).\)
4. Use technology to approximate \(P(-0.5\leq Z\leq 0.3).\)

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\(\int \frac{1}{1+x^{2}}dx\)

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\(\int {\sec x}\,dx\)

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\(\int e^{x^{1/3}}dx\)

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\(\int \ln (1+x)\,dx\)

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Even Functions Show that if \(f\) is an even function, then the Maclaurin expansion for \(f\) has only even powers of \(x.\)

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Odd Functions Show that if \(f\) is an odd function, then the Maclaurin expansion for \(f\) has only odd powers of \(x.\)

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Show that \((1+x)^{m}=\sum\limits_{k\,=\,0}^{\infty } {m\choose k}x^{k},\) when \(m\) is a nonnegative integer, by showing that \(R_{n}(x)\rightarrow 0\) as \(n\rightarrow \infty\).

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Show that the series \(\sum\limits_{k\,=\,0}^{\infty }{m\choose k}\,x^{k}\) converges absolutely for \(\vert x\vert <1\) and diverges for \(\vert x\vert >1\) if \(m<0\). (Hint: Use the Ratio Test.)

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Show that the interval of convergence of the Maclaurin expansion for \(f(x) =\sin ^{-1}x\) is \([-1,1].\)

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Euler’s Error Euler believed \(\frac{1}{2}=1-1+1-1+1-1\) \(+\cdots .\) He based his argument to support this equation on his belief in the identification of a series and the values of the function from which it was derived.

1. Write the Maclaurin expansion for \(\frac{1}{1+x}\). Do this without calculating any derivatives.
2. Evaluate both sides of the equation you derived in (a) at \(x=1\) to arrive at the formula above.
3. Criticize the procedure used in (b).

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Find the exact sum of the infinite series: \[\frac{x^{3}}{1(3)}-\frac{x^{5}}{3(5)}+\frac{x^{7}}{5(7)} - \frac{x^{9}}{7(9)}+\cdots \quad \hbox{for } x=1\]

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Find an elementary expression for \(\sum\limits_{k\,=\,1}^{\infty }\frac{x^{k+1}}{k(k+1) }\). Hint: Integrate the series for \(\ln \frac{1}{1-x}\).

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Show that \(\sum\limits_{k\,=\,1}^{\infty }\frac{k}{(k+1)!}=1\)

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Let \(s_{n}=\frac{1}{1!}+\frac{1}{2!}+\cdots \,+\frac{1}{n!},\quad n=1, 2, 3, \ldots \,\).

1. Show that \(n!\geq 2^{n-1}\).
2. Show that \(0<s_{n}\leq 1+\frac{1}{2}+\left( \frac{1}{2}\right) ^{2}+\cdots +\left( \frac{1}{2}\right) ^{n-1}\).
3. Show that \(0<s_{n}<s_{n+1}<2\). Then, conclude that \(S=\lim\limits_{n\,\rightarrow \,\infty }s_{n} \,\hbox{and}\, S\leq 2\).
4. Let \(t_{n}=\left[ 1+\frac{1}{n}\right] ^{n}\). Show that \[ \begin{eqnarray*} t_{n}&=&1+1+\frac{1}{2!}\left[ 1-\frac{1}{n}\right] + \frac{1}{3!}\left[ 1- \frac{1}{n}\right] \left[ 1-\frac{2}{n}\right] +\cdots \\[4pt] &&+\,\frac{1}{n!} \left[ 1-\frac{1}{n}\right] \left[ 1-\frac{2}{n}\right]\,\cdots \left[ 1-\frac{n-1}{n}\right] <s_{n}+1 \end{eqnarray*} \]
5. Show that \(0<t_{n}<t_{n+1}<3\). Then, conclude that \(e=\lim\limits_{n\,\rightarrow \,\infty }t_{n}\leq 3\).

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Show that \(\left[ 1+\frac{1}{n}\right] ^{n}<e\) for all \(n>0\).

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From the fact that \(\sin t\leq t\) for all \(t\geq 0\), use integration repeatedly to prove \[ 1-\frac{x^{2}}{2!}\leq \cos x\leq 1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}\qquad \hbox{for all }x\geq 0 \]

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Find the first four nonzero terms of the Maclaurin expansion for \(f(x) =(1+x) ^{x}.\)

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Show that \(f(x) =\left\{ \begin{array}{l@{\quad}l} e^{-1/x^{2}} & x\neq 0 \\ 0 & x=0 \end{array} \right.\) has a Maclaurin expansion at \(x=0.\) Then show that the Maclaurin series does not converge to \(f.\)