REVIEW EXERCISES

In Problems 1 and 2, the \(n\)th term of a sequence \(\{ s_{n}\}\) is given. Write the first five terms of each sequence.

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\(s_{n}=\dfrac{(-1) ^{n+1}}{n^{4}}\)

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\(s_{n}=\dfrac{2^{n}}{3^{n}}\)

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Find an expression for the \(n\)th term of the sequence, \(2,-\dfrac{3}{2},\,\dfrac{9}{8},\) \(-\dfrac{27}{32},\,\dfrac{81}{128}, \ldots ,\) assuming the indicated pattern continues for all \(n\).

In Problems 4 and 5, use properties of convergent sequences to find the limit of each sequence.

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\(\left\{ 1+\dfrac{n}{n^{2}+1}\right\}\)

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\(\left\{ \ln \dfrac{n+2}{n}\right\}\)

In Problems 6 and 7, use a related function or the Squeeze Theorem for sequences to show each sequence converges. Find its limit.

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\(\left\{\tan ^{-1}n\right\}\)

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\(\left\{\dfrac{(-1) ^{n}}{(n+1) ^{2}}\right\}\)

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Determine if the sequence \(\left\{\dfrac{e^{n}}{(n+2) ^{2}}\right\}\) is monotonic. If it is monotonic, is it increasing, nondecreasing, decreasing, or nonincreasing? Is it bounded from above and/or from below? Does it converge?

In Problems 9–12, determine whether each sequence converges or diverges. If it converges, find its limit.

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\(\{n!\}\)

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\(\left\{\left(\dfrac{5}{8}\right)^{n}\right\}\)

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\(\left\{ \left(-\dfrac{1}{2}\right)^{n}\right\}\)

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\(\left\{ (-1) ^{n}+e^{-n}\right\}\)

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Show that sequence \(\left\{ 1+\dfrac{2}{n}\right\}\) converges by showing it is either bounded from above and increasing or is bounded from below and decreasing.

633

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Find the fifth partial sum of \(\sum\limits_{k=1}^{\infty }\dfrac{(-1)^{k}}{4^{k-1}}.\)

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Find the sum of the telescoping series \(\sum\limits_{k=1}^{\infty }\left(\dfrac{4}{k+4}-\dfrac{4}{k+5}\right)\).

In Problems 16–19, determine whether each series converges or diverges. If it converges, find its sum.

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\(\sum\limits_{k=1}^{\infty}\dfrac{\cos ^{2}( k\pi ) }{k}\)

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\(\sum\limits_{k=1}^{\infty}-(\ln 2) ^{k}\)

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\(\sum\limits_{k=0}^{\infty}\dfrac{e}{3^{k}}\)

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\(\sum\limits_{k=1}^{\infty}(4^{1/3})^{k}\)

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Express \(0.123123123\ldots\) as a rational number using a geometric series.

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Show that the series \(\sum\limits_{k=1}^{\infty}\dfrac{3k-2}{k}\) diverges.

In Problems 22 and 23, use the Integral Test to determine whether each series converges or diverges.

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\(\sum\limits_{k=1}^{\infty}\dfrac{\ln k}{k^{2}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{1}{4k^{2}+9}\)

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Determine whether the \(p\)-series \(\sum\limits_{k=1}^{\infty}\dfrac{1}{k^{5/2}}\) converges or diverges. If it converges, find bounds for the sum.

In Problems 25–27, determine whether each series converges or diverges.

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\(\sum\limits_{k=5}^{\infty }\left[\dfrac{1}{k^{5}}\cdot \frac{1}{2^{k}}\right]\)

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\(\sum\limits_{k=1}^{\infty }\left[ \dfrac{3}{5^{k}}-\left( \dfrac{2}{3}\right) ^{k-1}\right] \)

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\(\sum\limits_{k=1}^{\infty }\dfrac{3}{k^{5}} \)

In Problems 28–30, use a Comparison Test to determine whether each series converges or diverges.

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\(\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{\sqrt{k+1}}\)

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\(\sum\limits_{k\,=\,1}^{\infty }\dfrac{k+1}{k^{k+1}}\)

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\(\sum\limits_{k\,=\,1}^{\infty }\dfrac{4}{k\,3^{k}}\)

In Problems 31–33, determine whether each alternating series converges or diverges. If the series converges, approximate the sum of each series correct to within 0.001.

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\(\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{k+2}{k(k+1) }\)

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\(\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{k^{2}}{e^{k}}\)

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\(\sum\limits_{k\,=\,1}^{\infty }(-1)^{k}\dfrac{3}{\sqrt[3]{k}}\)

In Problems 34–37, determine whether each series converges (absolutely or conditionally) or diverges.

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\(\sum\limits_{k=1}^{\infty}\sin \left( \dfrac{\pi}{2}k\right) \)

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\(\sum\limits_{k=1}^{\infty}\dfrac{(-1) ^{k+1}}{\sqrt{k}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{\cos k}{k^{3}}\)

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\(\dfrac{1}{2}-\dfrac{4}{2^{3}+1}+\dfrac{9}{3^{3}+1}-\dfrac{16}{4^{3}+1}+\cdots \)

In Problems 38 and 39, use the Ratio Test to determine whether each series converges or diverges.

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\(\sum\limits_{k=1}^{\infty } \dfrac{2^{k}}{k!}\)

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\(\sum\limits_{k\,=\,1}^{\infty }\dfrac{k!}{e^{k^{2}}}\)

In Problems 40 and 41, use the Root Test to determine whether each series converges or diverges.

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\(\sum\limits_{k=1}^{\infty }\dfrac{2^{k}}{(k+3)^{k+1}}\)

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\(\sum\limits_{k\,=\,1}^{\infty }(-1) ^{k} (e^{-k}-1) ^{k}\)

In Problems 42–52, determine whether each series converges or diverges.

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\(\sum\limits_{k=1}^{\infty}(-1) ^{k+1}\dfrac{2^{k+1}}{3^{k}}\)

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\(\sum\limits_{k=1}^{\infty}\ln \left( 1+\dfrac{1}{k}\right) \)

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\(\sum\limits_{k=5}^{\infty}\dfrac{3}{k\sqrt{k-4}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{1}{\left( 1+\dfrac{k^{2}+1}{k^{2}}\right) ^{k}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{2\cdot 4\cdot 6\cdots (2k)}{1\cdot 3\cdot 5\cdots (2k-1)}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{k^{2}}{(1+k^{3})\ln \sqrt[3]{1+k^{3}}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{k^{10}}{2^{k}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{\left(1+\dfrac{1}{k^{2}}\right)^{k^{2}}}{2^{k}}\)

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\(\sum\limits_{k\,=\,1}^{\infty}\left(\dfrac{k^{2}+1}{k}\right)^{k}\)

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\(\sum\limits_{k=1}^{\infty }\dfrac{k!}{3k^{k}}\)

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\(\sum\limits_{k\,=\,1}^{\infty }(-1)^{k+1}\dfrac{k+2}{3k-2}\)

In Problems 53–58,

  1. Find the radius of convergence of each power series.
  2. Find the interval of convergence of each power series.

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\(\sum\limits_{k=1}^{\infty}\dfrac{(x-3)^{3k-1}}{k^{2}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{x^{k}}{\sqrt[3]{k}}\)

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\(\sum\limits_{k=0}^{\infty}(-1)^{k}\dfrac{1}{k!(k+1)}\left( \dfrac{x}{2}\right) ^{2k+1}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{k^{k}}{(k!)^{2}}x^{k}\)

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\(\sum\limits_{k=1}^{\infty} \dfrac{(x-1)^k}{k} \)

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\(\sum\limits_{k=0}^{\infty}\dfrac{3^{k}x^{k}}{5^{k}}\)

In Problems 59 and 60, express each function as a power series centered at 0.

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\(f(x) =\dfrac{2}{x+3}\)

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

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  1. Use properties of a power series to find the power series representation for \(\int \dfrac{1}{1-3x^{2}}dx\).
  2. Use (a) to approximate \(\int_{0}^{1/2}\dfrac{1}{1-3x^{2}}dx\) correct to within 0.001.

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Find the Taylor expansion of \(f(x) =\dfrac{1}{1-2x}\) about \(c=1.\)

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Find the Taylor expansion of \(f(x) =e^{x/2}\) about \(c=1.\)

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Find the Maclaurin expansion of \(f(x)=2x^{3}-3x^{2}+x+5.\) Comment on the result.

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Find the Taylor expansion of \(f(x) =\tan x\) about \(c=\dfrac{\pi }{4}.\)

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Find the first five terms of the Maclaurin expansion for \(f(x) =e^{-x}\sin x\).

In Problems 67–69, use a binomial series to represent each function. Then determine its interval of convergence.

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\(f(x) =\dfrac{1}{\left( x+1\right) ^{4}}\)

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\(f(x) =\sqrt[3]{x^{2}-1}\)

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

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  1. Approximate \(y=\cos x\) by using the first four nonzero terms of the Taylor expansion for \(y=\cos x\) about \(\dfrac{\pi }{2.}\).
  2. Graph \(y=\cos x\) along with the Taylor expansion found in (a).
  3. Use (a) to approximate \(\cos 88 {{}^\circ}\).
  4. What is the error in using this approximation?
  5. If the approximation in (a) is used, for what values of \(x\) is the error less than \(0.0001?\)

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Use the Maclaurin expansion for \(f(x) =e^{x}\) to approximate \(e^{0.3}\) correct to three decimal points

In Problems 72 and 73, use properties of power series to approximate each integral using the first four terms of a Maclaurin series.

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\(\int_{0}^{1/2}\dfrac{dx}{\sqrt{1-x^{3}}}\)

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\(\int_{0}^{1/2} e^{x^{2}}dx\)