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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.

# Level 3

Calculate the sum of series below. $\frac{\pi^2}{3!}-\frac{\pi^4}{5!}+\frac{\pi^6}{7!}-\frac{\pi^8}{9!}+\cdots$

$1 + \frac{1+2}{2!} + \frac{1+2+2^2}{3!} + \frac{1+2+2^2+2^3}{4!} + \ldots$

Find the coefficient of $$x^2$$ in the Maclaurin series expansion for $\large (1+e^{x})^{10}.$

Evaluate the 25th derivative of $x^3\sin(x^2) \text{ at } x = 0.$

Hint: Start with the Maclaurin series for $$\sin(x)$$ to obtain a series for $$\sin(x^2)$$ and then for $$x^3 \sin(x^2).$$

$\large\dfrac1{1\cdot2}+\dfrac1{3\cdot4}+\dfrac1{5\cdot6}+\dfrac1{7\cdot8}+\ldots$

Let $$S$$ denote the value of series above. Find the value of $$e^S$$.

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