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Calculus
# Taylor Series

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Determine the Taylor series for the function

\[f(x)=\int _0 ^1 \dfrac{\sin x}{x}dx ~\text{ centered at }~ x=0.\]

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Evaluate \[\frac{\pi^2}{2!}-\frac{\pi^4}{4!}+\frac{\pi^6}{6!}-\frac{\pi^8}{8!}+\cdots.\]

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Determine the Taylor series for the function

\[f(x)=\sin x\cos x ~\text{ centered at }~ x=0.\]

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Evaluate

\[ \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n \cdot 2^n }. \]

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If \(f(x)=e^{x^2}\), what is \(f^{(2016)}(0)\) ?

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