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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

Determine the Taylor series for the function \[f(x) = \sin(x)\cos(x) \text{ centered at } x = \pi.\]

At \( a = 0\), what is the Taylor series expansion of

\[ \ln ( 1 + x ) ? \]

At \( a = 0\), what is the Taylor series expansion of

\[ \frac{1}{(1-x)^2} ? \]

Given the Maclaurin series expansion of \(\exp(x^2)\) as \[a_0 + a_1 x^1 + a_2 x^2 + \cdots ,\] what is the value of \(a_0 + a_1 + a_2 \)?

Determine the first three non-zero terms of the Taylor series for \[f(x) = \tan(x) \text{ centered at } x = \frac{\pi}{4}.\]

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