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Taylor Series
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.
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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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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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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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