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Chapter 9: Problem 71
Identify the two series that are the same. (a) \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !}\) (b) \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{(2 n-1) !}\) (c) \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{(2 n+1) !}\)
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Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=e^{x}, \quad c=1 $$
Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(1+\frac{2}{3}+\frac{3}{3^{2}}+\frac{4}{3^{3}}+\frac{5}{3^{4}}+\frac{6}{3^{5}}+\cdots\)
Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=e^{2 x}, \quad c=0 $$
Finding a Series Explain how to use the series $$g(x)=e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$$ to find the series for each function. Do not find the series. (a) \(f(x)=e^{-x}\) (b) \(f(x)=e^{3 x}\) (c) \(f(x)=x e^{x}\)
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 \(\left(\begin{array}{l}{k} \\ {0}\end{array}\right)=1\) $$ \left(\begin{array}{c}{-1 / 3} \\ {5}\end{array}\right) $$
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