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Magazine Problem 1
Determine the sums of the following geometric series when they are convergent. $$1+\frac{1}{6}+\frac{1}{6^{2}}+\frac{1}{6^{3}}+\frac{1}{6^{4}} \cdots$$
Problem 1
Determine the third Taylor polynomial of the given function at \(x=0.\) $$f(x)=\sin x$$
Problem 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.) $$\sum_{k=1}^{\infty} \frac{3}{\sqrt{k}}$$
Problem 1
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. $$\frac{1}{2 x+3}$$
Problem 2
Determine the sums of the following geometric series when they are convergent. $$1+\frac{3}{4}+\left(\frac{3}{4}\right)^{2}+\left(\frac{3}{4}\right)^{3}+\left(\frac{3}{4}\right)^{4}+\cdots$$
Problem 2
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. $$\ln (1-3 x)$$
Problem 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.) $$\sum_{k=1}^{\infty} \frac{5}{k^{3 / 2}}$$
Problem 2
Determine the third Taylor polynomial of the given function at \(x=0.\) $$f(x)=e^{-x / 2}$$
Problem 3
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. $$\sqrt{1+x}$$
Problem 3
Determine the sums of the following geometric series when they are convergent. $$1-\frac{1}{3^{2}}+\frac{1}{3^{4}}-\frac{1}{3^{6}}+\frac{1}{3^{8}}-\cdots$$
