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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.

# Taylor Series - Problem Solving

Determine the Taylor series for the function

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

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

Determine the Taylor series for the function

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

Evaluate

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

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

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