# Tricky Summation

Calculus Level 4

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.

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