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

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 \(2 .\) $$\frac{1}{1+x}$$

Determine the sums of the following geometric series when they are convergent. $$\frac{1}{5}+\frac{1}{5^{4}}+\frac{1}{5^{7}}+\frac{1}{5^{10}}+\frac{1}{5^{13}}+\cdots$$

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 \(2 .\) $$\frac{1}{1+x^{2}}$$

Determine the third Taylor polynomial of the given function at \(x=0\). $$f(x)=x e^{3 x}$$

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

Use three repetitions of the Newton-Raphson algorithm to approximate the following: The zero of \(\sin x+x^{2}-1\) near \(x_{0}=0\).

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 three repetitions of the Newton-Raphson algorithm to approximate the following: The zero of \(e^{x}+10 x-3\) near \(x_{0}=0\).

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