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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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by **Brilliant Staff**

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

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by **Brilliant Staff**

Determine the Taylor series for the function

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

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by **Brilliant Staff**

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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by **Brilliant Staff**

© Brilliant 2017

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