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Magazine Problem 29
Evaluate the given functions by using three terms of the appropriate Taylor series. $$\sin 61^{\circ}$$
Problem 29
Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent. $$10+9+8.1+7.29+6.561+\cdots$$
Problem 30
Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent. $$4+1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots$$
Problem 30
Evaluate the given functions by using three terms of the appropriate Taylor series. $$\cos \frac{7 \pi}{30}$$
Problem 30
Solve the given problems. Use long division to find a series expansion for \(f(x)=\frac{1}{(1+x)^{2}}\).
Problem 30
Solve the given problems by using series expansions. Evaluate \(\sin 32^{\circ}\) by first finding the expansion for \(\sin (x+\pi / 6)\)
Problem 31
Solve the given problems. Is it possible to find a Maclaurin expansion for (a) \(f(x)=\csc x\) or (b) \(f(x)=\ln x ?\) Explain.
Problem 31
Solve the given problems. Evaluate \(\int_{0}^{1} e^{x} d x\) directly and compare the result obtained by using four terms of the series for \(e^{x}\) and then integrating.
Problem 31
Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent. $$512-64+8-1+\frac{1}{8}-\dots$$
Problem 31
Solve the given problems by using series expansions. We can evaluate \(\pi\) by use of \(\frac{1}{4} \pi=\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}\) (see Exer- cise 62 of Section 20.6 ), along with the series for \(\tan ^{-1} x .\) The first three terms are \(\tan ^{-1} x=x-\frac{1}{3} x^{3}+\frac{1}{5} x^{5} .\) Using these terms, expand \(\tan ^{-1} \frac{1}{2}\) and \(\tan ^{-1} \frac{1}{3}\) and approximate the value of \(\pi\)
