91Ó°ÊÓ

Determine the sums of the following geometric series when they are convergent. $$1-\frac{1}{3^{2}}+\frac{1}{3^{4}}-\frac{1}{3^{6}}+\frac{1}{3^{8}}-\cdots$$

Sum an appropriate infinite series to find the rational number whose decimal expansion is given. $$.1515 \overline{15}$$

Estimating the Root of a Funtion Suppose that the graph of the function \(f(x)\) has slope \(-2\) at the point \((1,2)\). If the Newton-Raphson algorithm is used to find a root of \(f(x)=0\) with the initial guess \(x_{0}=1\), what is \(x_{1} ?\)

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

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 \(x=0\).

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?

Total Distance Traveled by a Bouncing Ball The coefficient of restitution of a ball, a number between 0 and 1 , specifies how much energy is conserved when the ball hits a rigid surface. A coefficient of \(.9\), for instance, means a bouncing ball will rise to \(90 \%\) of its previous height after each bounce. The coefficients of restitution for a tennis ball, basketball, super ball, and softball are \(.7, .75, .9\), and \(.3\), respectively. Find the total distance traveled by a tennis ball dropped from a height of 6 feet.

Determine the sums of the following infinite series: $$\sum_{k=1}^{\infty}\left(\frac{1}{3}\right)^{2 k}$$