91Ó°ÊÓ

Determine the third Taylor polynomial of the given function at \(x=0.\) $$f(x)=5 e^{2 x}$$

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.) $$\sum_{k=0}^{\infty} \frac{7}{k+100}$$

Find the Taylor series at \(x=0\) of the given function by computing three or four derivatives and using the definition of the Taylor series. $$(1+x)^{3}$$

Determine the third Taylor polynomial of the given function at \(x=0.\) $$f(x)=\cos (\pi-5 x)$$

Determine the sums of the following geometric series when they are convergent. $$1+\frac{1}{2^{3}}+\frac{1}{2^{6}}+\frac{1}{2^{9}}+\frac{1}{2^{12}}+\cdots$$

Determine the third Taylor polynomial of the given function at \(x=0.\) $$f(x)=\sqrt{4 x+1}$$

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. $$\frac{1}{1-3 x}$$

Use three repetitions of the Newton-Raphson algorithm to approximate the following: The zero of \(x^{2}-x-5\) between 2 and 3

Determine the sums of the following geometric series when they are convergent. $$2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}+\cdots$$

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.) $$\sum_{k=1}^{\infty} \frac{2}{5 k-1}$$