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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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