91Ó°ÊÓ

Determine the sums of the following geometric series when they are convergent. $$\frac{1}{3^{2}}-\frac{1}{3^{3}}+\frac{1}{3^{4}}-\frac{1}{3^{5}}+\frac{1}{3^{6}}-\cdots$$

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.) $$\sum_{k=1}^{\infty} \frac{1}{(2 k+1)^{3}}$$

Determine the fourth Taylor polynomial of \(f(x)=e^{x}\) at \(x=0,\) and use it to estimate \(e^{0.01}.\)

Determine the sums of the following geometric series when they are convergent. $$3-\frac{3^{2}}{7}+\frac{3^{3}}{7^{2}}-\frac{3^{4}}{7^{3}}+\frac{3^{5}}{7^{4}}-\dots$$

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.) $$\sum_{k=1}^{\infty} \frac{1}{(3 k)^{2}}$$

Determine the fourth Taylor polynomial of \(f(x)=\ln (1-x)\) at \(x=0,\) and use it to estimate \(\ln (.9).\)

Determine the sums of the following geometric series when they are convergent. $$6-1.2+.24-.048+.0096-\cdots$$

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

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. $$5 e^{x / 3}$$

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