Chapter 12

Expand \(f(x)\) in powers of \(x,\) basing your calculations on the geometric series $$\frac{1}{1-x}=1+x+x^{2}+\cdots+x^{n}+\cdots$$ $$f(x)=\frac{1}{(1-x)^{2}}$$

Suppose that the series \(\sum_{k=0}^{\infty} a_{k} x^{k}\) converges at \(x=3 .\) What can you conclude about the convergence or divergence of the following series? (a) \(\sum_{k=0}^{\infty} a_{k} 2^{k}\) (b) \(\sum_{k=0}^{\infty} a_{k}(-2)^{k}\) (c) \(\sum_{k=0}^{\infty} a_{k}(-3)^{k}\) (d) \(\sum_{k=0}^{\infty} a_{k} 4^{k}\)

Find the Taylor polynomial of the function \(f\) for the given values of \(a\) and \(n\) and give the Lagrange form of the remainder. $$f(x)=\sqrt{x} ; \quad a=4, \quad n=3.$$

Find the Taylor polynomial \(P_{4}\) for the function \(f\) $$f(x)=x-\cos x$$

Determine whether the series converges or diverse. $$\sum \frac{k}{k^{3}+1}$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(1+(-1)+1+\dots+(-1)^{k}+\cdots\)

Find the sum of the series. $$\sum_{k=1}^{\infty} \frac{1}{2 k(k+1)}$$

Evaluate. $$\sum_{k=0}^{2}(3 k+1)$$

Expand \(f(x)\) in powers of \(x,\) basing your calculations on the geometric series $$\frac{1}{1-x}=1+x+x^{2}+\cdots+x^{n}+\cdots$$ $$f(x)=\frac{1}{(1-x)^{3}}$$

Suppose that the series \(\sum_{k=0}^{\infty} a_{k} x^{k}\) converges at \(x=-3\) and diverges at \(x=5 .\) What can you conclude about the convergence or divergence of the following series? (a) \sum_{k=0}^{\infty} a_{k} 2^{k} (b) \(\sum_{k=0}^{\infty} a_{k}(-6)^{k}\) (c) \(\sum_{k=0}^{\infty} a_{k} 4^{k}\) (d) \(\sum_{k=0}^{\infty}(-1)^{k} a_{k} 3^{k}\)

Find the Taylor polynomial \(P_{4}\) for the function \(f\) $$f(x)=\sqrt{1+x}$$

Find the Taylor polynomial of the function \(f\) for the given values of \(a\) and \(n\) and give the Lagrange form of the remainder. $$f(x)=\cos x ; \quad a=\pi / 3, \quad n=4.$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\frac{1}{4}-\frac{1}{6}+\frac{1}{8}-\frac{1}{10}+\dots+\frac{(-1)^{k}}{2 k}+\cdots\)

Determine whether the series converges or diverges. $$\sum \frac{1}{k 2^{k}}$$

Find the sum of the series. $$\sum_{k=3}^{\infty} \frac{1}{k^{2}-k}$$

Evaluate. $$\sum_{k=1}^{4}(3 k-1)$$

Determine whether the series converges or diverse. $$\sum \frac{1}{3 k+2}$$

Expand \(f(x)\) in powers of \(x,\) basing your calculations on the geometric series $$\frac{1}{1-x}=1+x+x^{2}+\cdots+x^{n}+\cdots$$ $$f(x)=\frac{1}{(1-x)^{k}}$$

Find the Taylor polynomial \(P_{4}\) for the function \(f\) $$f(x)=\ln \cos x$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+\dots+(-1)^{k+1} \frac{k}{(k+1)}+\cdots\).

Determine whether the series converges or diverges. $$\sum \frac{1}{k^{k}}$$

Determine whether the series converges or diverse. $$\sum \frac{1}{(2 k+1)^{2}}$$

Find the sum of the series. $$\sum_{k=1}^{\infty} \frac{1}{k(k+3)}$$

Evaluate. $$\sum_{k=0}^{3} 2^{k}$$

Find the interval of convergence. $$\sum k x^{k}$$

Expand \(f(x)\) in powers of \(x,\) basing your calculations on the geometric series $$\frac{1}{1-x}=1+x+x^{2}+\cdots+x^{n}+\cdots$$ $$f(x)=\ln (1-x)$$

Find the interval of convergence. $$\sum \frac{1}{k} x^{k}$$

Find the Taylor polynomial of the function \(f\) for the given values of \(a\) and \(n\) and give the Lagrange form of the remainder. $$f(x)=\ln x ; \quad a=1, \quad n=5.$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\frac{1}{2 \ln 2}-\frac{1}{3 \ln 3}+\frac{1}{4 \ln 4}-\frac{1}{5 \ln 5}+\cdots+(-1)^{k} \frac{1}{k \ln k}+\cdots\).

Determine whether the series converges or diverges. $$\sum\left(\frac{k}{2 k+1}\right)^{k}$$

Evaluate. $$\sum_{k=1}^{4} \frac{1}{2^{k}}$$

Find the Taylor polynomial \(P_{4}\) for the function \(f\) $$f(x)=\sec x$$

Determine whether the series converges or diverse. $$\sum \frac{\ln k}{k}$$

Expand \(f(x)\) in powers of \(x,\) basing your calculations on the geometric series $$\frac{1}{1-x}=1+x+x^{2}+\cdots+x^{n}+\cdots$$ $$f(x)=\ln \left(1-x^{2}\right)$$

Find the Taylor polynomial of the function \(f\) for the given values of \(a\) and \(n\) and give the Lagrange form of the remainder. $$f(x)=\arctan x: \quad a=1, \quad n=3.$$

Find the Taylor polynomial \(P_{5}\) for the given function \(f\). $$f(x)=(1+x)^{-1}$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum(-1)^{k} \frac{\ln k}{k}\).

Determine whether the series converges or diverges. $$\sum \frac{k !}{100^{k}}$$

Find the sum of the series. $$\sum_{k=0}^{\infty} \frac{3}{10^{k}}$$

Evaluate. $$\sum_{k=0}^{3}(-1)^{k} 2^{k}$$

Determine whether the series converges or diverse. $$\sum \frac{1}{\sqrt{k+1}}$$

Expand \(f(x)\) in powers of \(x,\) basing your calculations on the geometric series $$\frac{1}{1-x}=1+x+x^{2}+\cdots+x^{n}+\cdots$$ $$f(x)=\ln (2-3 x)$$

Find the interval of convergence. $$\sum \frac{2^{k}}{k^{2}} x^{k}$$

Find the Taylor polynomial \(P_{5}\) for the given function \(f\). $$f(x)=e^{x} \sin x$$

Find the Taylor polynomial of the function \(f\) for the given values of \(a\) and \(n\) and give the Lagrange form of the remainder. $$f(x)=\cos \pi x ; \quad a=\frac{1}{2}, \quad n=4.$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum(-1)^{k} \frac{k}{\ln k}\).

Determine whether the series converges or diverges. $$\sum \frac{(\ln k)^{2}}{k}$$

Determine whether the series converges or diverse. $$\sum \frac{1}{k^{2}+1}$$

Find the sum of the series. $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{5^{k}}$$

Find the interval of convergence. $$\sum(-k)^{2 k} x^{2 k}$$