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True or False Every power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}(x-c) ^{k}\) converges for at least one number.

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True or False Let \(b_{n}\) denote the \(n\)th term of the power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\). If \( \lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{b_{n+1}}{b_{n}} \right\vert \lt 1\) for every number \(x\), then \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) is absolutely convergent on the interval \(( -\infty ,\,\infty ) \).

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True or False If the radius of convergence of a power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) is \(0\), then the power series converges only for \(x=0\).

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True or False If a power series converges at one endpoint of its interval of convergence, then it must converge at its other endpoint.

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True or False The power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) and \(\sum\limits_{k\,=\,0}^{\infty }a_{k}(x-3)^{k}\) have the same radius of convergence.

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True or False The power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) and \(\sum\limits_{k\,=\,0}^{\infty }a_{k}(x-3)^{k}\) have the same interval of convergence.

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True or False If the power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) converges for \(x=8\), then it converges for \(x=-8\).

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True or False If the power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) converges for \(x=3\), then it converges for \(x=1\).

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True or False If the power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) converges for \(x=-4\), then it converges for \(x=3\).

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True or False If the power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) converges for \(x=3\), then it diverges for \(x=5\).

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True or False A possible interval of convergence for the power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) is \([-2,4]\).

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True or False If the power series \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) diverges for a number \(x_{1}\), then it converges for all numbers \(x\) for which \(\vert x\vert <\vert x_{1}\vert \).

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\(\sum\limits_{k\,=\,0}^{\infty} kx^{k}\)

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

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\(\sum\limits_{k\,=\,0}^{\infty }\dfrac{(x+1)^{k}}{3^{k}}\)

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

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{x^{k}}{2^{k}(k+1) }\)

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

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

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{x^{k}}{1+k^{2}}\)

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{k^{2}x^{k}}{3^{k}}\)

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{2^{k}x^{k}}{3^{k}}\)

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{kx^{k}}{2k+1}\)

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\(\sum\limits_{k\,=\,0}^{\infty}(6x)^{k}\)

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\(\sum\limits_{k\,=\,0}^{\infty}(x-3)^{k}\)

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{k(2x)^{k}}{3^{k}}\)

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

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

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

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{k(x-2) ^{k}}{3^{k}}\)

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

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\(\sum\limits_{k=1}^{\infty}(kx)^{k}\)

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

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

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\(\sum\limits_{k=0}^{\infty}\dfrac{k(k+1)x^{k}}{4^{k}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{(-1)^{k}(x-5)^{k}}{k(k+1)}\)

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\(\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{(x-3)^{2k}}{9^{k}}\)

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

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

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{(x+1)^{k}}{k!}\)

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\(\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{(x-1)^{4k}}{k!}\)

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

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

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{3^{k}(x-2)^{k}}{k!}\)

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A function \(f\) is defined by the power series \(f(x) =\sum\limits_{k\,=\,0}^{\infty }\dfrac{x^{k}}{3^{k}}\).

1. Find the domain of \(f\).
2. Evaluate \(f(2)\) and \(f(-1)\).
3. Find \(f\).

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A function \(f\) is defined by the power series \(f(x)=\sum\limits_{k\,=\,0}^{\infty }(-1) ^{k}\left(\dfrac{x}{2}\right)^{k}\).

1. Find the domain of \(f\).
2. Evaluate \(f(0)\) and \(f(1)\).
3. Find \(f\).

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A function \(f\) is defined by the power series \(f(x)=\sum\limits_{k\,=\,0}^{\infty }\dfrac{(x-2) ^{k}}{2^{k}}\).

1. Find the domain of \(f\).
2. Evaluate \(f(1)\) and \(f(2)\).
3. Find \(f\).

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A function \(f\) is defined by the power series \(f(x) =\sum\limits_{k\,=\,0}^{\infty }(-1) ^{k}(x+3)^{k}\).

1. Find the domain of \(f\).
2. Evaluate \(f(-3)\) and \(f(-2.5)\).
3. Find \(f\).

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If \(\sum\limits_{k=0}^{\infty }a_{k\,}x^{k}\) converges for \(x=3\), what, if anything, can be said about the convergence at \(x=2\)? Can anything be said about the convergence at \(x=5\)?

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If \(\sum\limits_{k=0}^{\infty }a_{k}(x-2)^{k}\) converges for \(x=6\), at what other numbers \(x\) must the series necessarily converge?

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If the series \(\sum\limits_{k=0}^{\infty }\,a_{k}x^{k}\) converges for \(x=6\) and diverges for \(x=-8\), what, if anything, can be said about the truth of the following statements?

1. The series converges for \(x=2\).
2. The series diverges for \(x=7\).
3. The series is absolutely convergent for \(x=6\).
4. The series converges for \(x=-6\).
5. The series diverges for \(x=10\).
6. The series is absolutely convergent for \(x=4\).

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If the radius of convergence of the power series \(\sum\limits_{k=0}^{\infty }\,a_{k}(x-3) ^{k}\) is \(R=5\), what, if anything, can be said about the truth of the following statements?

1. The series converges for \(x=2\).
2. The series diverges for \(x=7\).
3. The series diverges for \(x=8\).
4. The series converges for \(x=-6\).
5. The series converges for \(x=-2\).

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

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\(f(x) =\dfrac{1}{1-x^{2}}\)

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\(f(x) =\dfrac{1}{6-2x}\)

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\(f(x) =\dfrac{4}{x+2}\)

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

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\(f(x) =\dfrac{4x^{2}}{x+2}\)

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

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\(f(x)=\sum\limits_{k\,=\,0}^{\infty}\dfrac{(-1)^{k}x^{2k}}{(2k) !}\)

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\(f(x)=\sum\limits_{k\,=\,0}^{\infty}\dfrac{x^{k}}{k!}\)

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\(f(x)=\sum\limits_{k\,=\,0}^{\infty}\dfrac{(-1)^{k}x^{k}}{k!}\)

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

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

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\(f(x)=\dfrac{2}{3 (1-x) ^{2}} \)

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\(f(x) =\dfrac{1}{(1-x) ^{4}}\)

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

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\(f(x) =\ln (1-2x) \)

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\(f(x) =\ln (1-x^{2})\)

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

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

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\(\sum\limits_{k=1}^{\infty}\dfrac{(x-4)^{k}}{k}\)

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

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

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\(\sum\limits_{k\,=\,0}^{\infty}x^{k^{2}}\)

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\(\sum\limits_{k=1}^{\infty}\dfrac{k^{a}}{a^{k}}(x-a)^{k}, \quad a\neq 0\)

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

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\(\sum\limits_{k\,=\,0}^{\infty}\dfrac{\sqrt{k!}}{(2k)!}x^{k}\)

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1. In the geometric series \(\dfrac{1}{1-x}=\sum \limits_{k\,=\,0}^{\infty }x^{k}\), \(-1\lt x \lt 1\), replace \(x\) by \(x^{2}\) to obtain the power series representation for \(\dfrac{1}{1-x^{2}}\).
2. What is its interval of convergence?

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1. Integrate the power series found in Problem 79 for \(\dfrac{1}{1-x^{2}}\) to obtain the power series \(\dfrac{1}{2}\ln \dfrac{1+x}{1-x}\).
2. What is its interval of convergence?

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Use the power series found in Problem 80 to get an approximation for \(\ln 2\) correct to three decimal places.

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Use the first 1000 terms of Gregory’s series to approximate \(\dfrac{\pi }{4}\). What is the approximation for \(\pi ?\)

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If \(R>0\) is the radius of convergence of \(\sum\limits_{k=1}^{\infty }a_{k}x^{k}\), show that \(\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert =\dfrac{1}{R}\), provided this limit exists.

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If \(R\) is the radius of convergence of \(\sum\limits_{k=1}^{\infty }a_{k}x^{k}\), show that the radius of convergence of \(\sum\limits_{k=1}^{\infty }a_{k}x^{2k}\) is \(\sqrt{R}\).

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Prove that if a power series is absolutely convergent at one endpoint of its interval of convergence, then the power series is absolutely convergent at the other endpoint.

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Suppose \(\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) converges for \(\vert x\vert <R\) and that \(\lim\limits_{n\rightarrow \infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert\) exists. Show that \(\sum\limits_{k=1}^{\infty }ka_{k}x^{k-1}\) and \(\sum\limits_{k\,=\,0}^{\infty } \dfrac{a_{k}}{k+1}x^{k+1}\) also converge for \(\vert x\vert <R\).

Question

Consider the differential equation \[ (1+x^{2})\,y^{\prime \prime} -4xy^\prime +6y=0 \]

Assuming there is a solution \(y(x)=\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) , substitute and obtain a formula for \(a_{k}\). Your answer should have the form \[ y(x)=a_{0}(1-3x^{2}) +a_{1}\!\left( x-\dfrac{1}{3}x^{3}\right)\qquad a_{0},~a_{1} \hbox{ real numbers} \]

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If the series \(\sum\limits_{k=0}^{\infty }a_{k}3^{k}\) converges, show that the series \(\sum\limits_{k=1}^{\infty }ka_{k}2^{k}\) also converges.

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Find the interval of convergence of the series \(\sum\limits_{k=1}^{\infty }\dfrac{(x-2)^{k}}{k(3^{k})}\).

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Let a power series \(S(x)\) be convergent for \(\vert x\vert <R\). Assume that \(S(x)=\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}\) with partial sums \(S_{n}(x)=\sum\limits_{k\,=\,0}^{n}a_{k}x^{k}\). Suppose for any number \(\varepsilon >0\), there is a number \(N\) so that when \(n>N\), \(\vert S(x)-S_{n}(x) \vert <\dfrac{\varepsilon }{3}\) for all \(\vert x\vert <R\). Show that \(S(x) \) is continuous for all \(\vert x\vert <R\).

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Find the power series in \(x\), denoted by \(f(x)\), for which \(f^{\prime \prime} (x) + f(x) =0\) and \(f(0) =0\), \(f^\prime (0) =1\). What is the radius of convergence of the series?

Question

The Bessel function of order \(m\) of the first kind, where \(m\) is a nonnegative integer, is defined as \[ J_{m}(x) =\sum\limits_{k\,=\,0}^{\infty }(-1) ^{k} \dfrac{1}{(k+m) !\, k!}\left( \dfrac{x}{2}\right) ^{2k+m} \]

Show that:

1. \(J_{0}(x) =x^{-1}\dfrac{d}{dx}(xJ_{1}(x) )\)
2. \(J_{1}(x) =x^{-2}\dfrac{d}{dx}(x^{2}J_{2}(x) )\)