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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: Level 5 Challenges

$f\left( x \right) =\sum _{ n=0 }^{ \infty }{ \frac { \sin { \left( xn \right) } }{ { 7 }^{ n } } }$ For all real numbers $$x$$, let $$f(x)$$ be a function with fundamental period $$P$$.

Let $$\displaystyle a = \int_0^{P/2} f(x) \, dx$$. If $$f \left( \pi e^{|a|} \right)$$ can be expressed as $$-\dfrac{\alpha\sqrt \beta}{\gamma}$$, where $$\alpha, \beta$$ and $$\gamma$$ are positive integers with $$\alpha, \gamma$$ coprime and $$\beta$$ square-free, find $$\alpha + \beta + \gamma$$.

$\frac {1^7}{1!} + \frac {1^7 + 2^7}{2!} + \frac {1^7 + 2^7 + 3^7}{3!} + \frac {1^7 + 2^7 + 3^7 + 4^7}{4!} + \ldots$

If the series above equals to $$W$$, what is the value of $$\frac {24}{e} \times W$$?

Note: $$e = \displaystyle \lim_{n \to \infty} \left (1 + \frac 1 n \right )^n$$

Inspired by Caleb Townsend

Image Credit: Flickr Gingertwist

For all integers $$n$$, we define $$\xi_n$$ as follows: $\begin{cases} \xi_n = 1 & \text{if } n \equiv 0 \pmod{4} \text{ or } n \equiv 1 \pmod{4} \\ \xi_n= -1 & \text{if } n \equiv 2 \pmod{4} \text{ or } n \equiv 3 \pmod{4} \end{cases}$ For all $$n \in \mathbb{Z^+}$$, let $f(n)= \xi_0 \dbinom{n}{0} + \xi_1 \dbinom{n}{1} + \xi_2 \dbinom{n}{2} + \cdots + \xi_n \dbinom{n}{n}.$ Find $$\left \lfloor 100 \left( \displaystyle \sum \limits_{n=0}^{\infty} \dfrac{f(n)}{n!} \right) \right \rfloor$$.

Details and assumptions

• As an explicit example, since $$4 \equiv 0 \pmod{4}$$, $$\xi_4= 1$$, whereas $$\xi_6 = -1$$ since $$6 \equiv 2 \pmod{4}$$. Note that $$\xi_0= \xi_1= 1$$.

• The floor function $$\lfloor x \rfloor$$ denotes the largest integer less than or equal to $$x$$. For example, $$\lfloor 3.25 \rfloor = 3, \lfloor 4 \rfloor= 4, \lfloor \pi \rfloor = 3$$.

• You might use a scientific calculator for this problem.

$\large\displaystyle \lim_{x \to 0} \frac {(\cos x)^{\sin x} - \sqrt{1-x^3}}{x^6} =\ ?$

Find $$\dfrac{f^{(2016)}(0)}{2016!}$$ for $$f(x)=\dfrac{x^3}{(1-x)^3(1+x+x^2)}$$.

Clarifications:

• $$f^{(k)}(x)$$ denotes the $$k^\text{th}$$ derivative of $$f(x)$$.

• $$!$$ denotes the factorial notation. For example, $$8! = 1\times2\times3\times\cdots\times8$$.

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