Calculus

Taylor Series

Taylor Series: Level 3 Challenges

         

Calculate the sum of series below. π23!π45!+π67!π89!+\frac{\pi^2}{3!}-\frac{\pi^4}{5!}+\frac{\pi^6}{7!}-\frac{\pi^8}{9!}+\cdots

1+1+22!+1+2+223!+1+2+22+234!+1 + \frac{1+2}{2!} + \frac{1+2+2^2}{3!} + \frac{1+2+2^2+2^3}{4!} + \ldots

Find the coefficient of x2x^2 in the Maclaurin series expansion for (1+ex)10.\large (1+e^{x})^{10}.

Evaluate the 25th derivative of x3sin(x2) at x=0.x^3\sin(x^2) \text{ at } x = 0.

Hint: Start with the Maclaurin series for sin(x)\sin(x) to obtain a series for sin(x2)\sin(x^2) and then for x3sin(x2).x^3 \sin(x^2).

112+134+156+178+\large\dfrac1{1\cdot2}+\dfrac1{3\cdot4}+\dfrac1{5\cdot6}+\dfrac1{7\cdot8}+\ldots

Let SS denote the value of series above. Find the value of eSe^S.

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