91Ó°ÊÓ

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.) $$\sum_{k=1}^{\infty} \frac{1}{e^{2 k+1}}$$

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.) $$\sum_{k=1}^{\infty} k e^{-k^{2}}$$

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. $$1-e^{-x}$$

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

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.) $$\sum_{k=1}^{\infty} k^{-3 / 4}$$

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.

Determine the sums of the following geometric series when they are convergent. $$\frac{5^{3}}{3}-\frac{5^{5}}{3^{4}}+\frac{5^{7}}{3^{7}}-\frac{5^{9}}{3^{10}}+\frac{5^{11}}{3^{13}}-\dots$$

Determine all Taylor polynomials for \(f(x)=x^{2}+2 x+1\) at \(x=0.\)

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.) $$\sum_{k=1}^{\infty} \frac{2 k+1}{k^{2}+k+2}$$

Sum an appropriate infinite series to find the rational number whose decimal expansion is given. $$.2727 \overline{27}$$