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Chapter 9: Problem 56
State the Alternating Series Test.
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Evaluating a Binomial Coefficient In Exercises \(89-92\) , evaluate the binomial coefficient using the formula $$\left(\begin{array}{l}{k} \\ {n}\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdot \cdot(k-n+1)}{n !}$$ where \(k\) is a real number, \(n\) is a positive integer, and \(\left(\begin{array}{l}{k} \\ {0}\end{array}\right)=1\) $$ \left(\begin{array}{l}{5} \\ {3}\end{array}\right) $$
Finding a Taylor Polynomial Using Technology In Exercises \(75-78\) , use a computer algebra system to find the fifth-degree Taylor polynomial, centered at \(c\) , for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. $$ f(x)=x \cos 2 x, \quad c=0 $$
Finding a Maclaurin Series In Exercises \(41-44\) , find the Maclaurin series for the function. (See Examples 7 and \(8 . )\) $$ h(x)=x \cos x $$
Finding a Limit In Exercises \(59-62,\) use the series representation of the function \(f\) to find \(\lim _{x \rightarrow 0} f(x)\) (if it exists). $$ f(x)=\frac{\sin x}{x} $$
Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(\frac{1}{(\ln 3)^{3}}+\frac{1}{(\ln 4)^{4}}+\frac{1}{(\ln 5)^{5}}+\frac{1}{(\ln 6)^{6}}+\cdots\)
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