91Ó°ÊÓ

Show that \(.99 \overline{9}=1\).

Use the comparison test to determine whether the infinite series is convergent or divergent. \(\sum_{k=2}^{\infty} \frac{1}{k^{2}+5}\left[\right.\) Compare with \(\left.\sum_{k=2}^{\infty} \frac{1}{k^{2}} .\right]\)

Find the Taylor series of \(x e^{x^{2}}\) at \(x=0\).

Use the second Taylor polynomial of \(f(x)=\sqrt{x}\) at \(x=9\) to estimate \(\sqrt{9.3}\).

Use the comparison test to determine whether the infinite series is convergent or divergent. \(\sum_{k=2}^{\infty} \frac{1}{\sqrt{k^{2}-1}}\left[\right.\) Compare with \(\left.\sum_{k=2}^{\infty} \frac{1}{k} .\right]\)

The hyperbolic cosine of \(x\), denoted by \(\cosh x\), is defined by $$ \cosh x=\frac{1}{2}\left(e^{x}+e^{-x}\right) . $$ This function occurs often in physics and probability theory. The graph of \(y=\cosh x\) is called a catenary. (a) Use differentiation and the definition of a Taylor series to compute the first four nonzero terms in the Taylor series of \(\cosh x\) at \(x=0\). (b) Use the known Taylor series for \(e^{x}\) to obtain the Taylor series for \(\cosh x\) at \(x=0\).

Use the comparison test to determine whether the infinite series is convergent or divergent. \(\sum_{k=1}^{\infty} \frac{1}{2^{k}+k}\left[\right.\) Compare with \(\left.\sum_{k=1}^{\infty} \frac{1}{2^{k}} .\right]\)

What special occurrence takes place when the NewtonRaphson algorithm is applied to the linear function \(f(x)=m x+b\) with \(m \neq 0 ?\)

The Multiplier Effect Compute the effect of a $$\$ 20$$ -billion federal income tax cut when the population's marginal propensity to consume is \(98 \%\). What is the "multiplier" in this case?

Determine the \(n\) th Taylor polynomial of \(f(x)=1 / x\) at \(x=1\).