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The Taylor series is a polynomial of infinite degree used to represent functions like sine, cube roots, and the exponential function. They're how some calculators (and Physicists) make approximations.

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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