91Ó°ÊÓ

Sketch the graphs of \(f(x)=\frac{1}{1-x}\) and its first three Taylor polynomials at \(x=0\).

Determine the sums of the following geometric series when they are convergent. $$\frac{2}{5^{4}}-\frac{2^{4}}{5^{5}}+\frac{2^{7}}{5^{6}}-\frac{2^{10}}{5^{7}}+\frac{2^{13}}{5^{8}}-\cdots$$

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) $$\sum_{k=1}^{\infty} \frac{1}{e^{2 k+1}}$$

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 \(2 .\) $$x^{3} e^{x^{2}}$$

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) $$\sum_{k=1}^{\infty} k e^{-k^{2}}$$

Determine the \(n\) th Taylor polynomial for \(f(x)=e^{x}\) at \(x=0 .\)

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 \(2 .\) $$1-e^{-x}$$

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 \(2 .\) $$3\left(e^{-2 x}-2\right)$$

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) $$\sum_{k=1}^{\infty} k^{-3 / 4}$$

Internal Rate of Return An investor buys a bond for $$\$ 1000$$. She receives $$$ 10$$ at the end of each month for 2 months and then sells the bond at the end of the second month for $$\$ 1040 .$$ Determine the internal rate of return on this investment.