91Ó°ÊÓ

Find the Taylor series at \(x=0\) of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at \(x=0\) of \(\frac{1}{1-x}, e^{x}\), or \(\cos x .\) These series are derived in Examples 1 and 2 and Check Your Understanding Problem \(2 .\) $$\cos x^{2}$$

Use the integral test to determine if \(\sum_{k=1}^{\infty} \frac{e^{1 / k}}{k^{2}}\) is convergent. Show that the hypotheses of the integral test are satisfied.

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

Sum an appropriate infinite series to find the rational number whose decimal expansion is given. $$4.011 \overline{011}(=4+.011 \overline{011})$$

It can be shown that \(\lim _{b \rightarrow \infty} b e^{-b}=0 .\) Use this fact and the integral test to show that \(\sum_{k=1}^{\infty} \frac{k}{e^{k}}\) is convergent.

Find the Taylor series at \(x=0\) of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at \(x=0\) of \(\frac{1}{1-x}, e^{x}\), or \(\cos x .\) These series are derived in Examples 1 and 2 and Check Your Understanding Problem \(2 .\) $$x \sin x^{2}$$

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 \(x_{1} ?\)

Determine the third and fourth Taylor polynomials of \(x^{3}+3 x-1\) at \(x=-1\).

Sum an appropriate infinite series to find the rational number whose decimal expansion is given. $$5.44 \overline{4}$$

Is the series \(\sum_{k=1}^{\infty} \frac{3^{k}}{4^{k}}\) convergent? What is the easiest way to answer this question? Can you tell if $$ \int_{1}^{\infty} \frac{3^{x}}{4^{x}} d x $$ is convergent?