Calculus

Taylor Series

Taylor Series: Level 5 Challenges

         

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

Let a=0P/2f(x)dx \displaystyle a = \int_0^{P/2} f(x) \, dx . If f(πea)f \left( \pi e^{|a|} \right) can be expressed as αβγ -\dfrac{\alpha\sqrt \beta}{\gamma}, where α,β\alpha, \beta and γ\gamma are positive integers with α,γ\alpha, \gamma coprime and β\beta square-free, find α+β+γ\alpha + \beta + \gamma.

171!+17+272!+17+27+373!+17+27+37+474!+ \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 WW, what is the value of 24e×W\frac {24}{e} \times W ?

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

Inspired by Caleb Townsend

Image Credit: Flickr Gingertwist

For all integers nn, we define ξn\xi_n as follows: {ξn=1if n0(mod4) or n1(mod4)ξn=1if n2(mod4) or n3(mod4)\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 nZ+n \in \mathbb{Z^+}, let f(n)=ξ0(n0)+ξ1(n1)+ξ2(n2)++ξn(nn).f(n)= \xi_0 \dbinom{n}{0} + \xi_1 \dbinom{n}{1} + \xi_2 \dbinom{n}{2} + \cdots + \xi_n \dbinom{n}{n}. Find 100(n=0f(n)n!)\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 40(mod4)4 \equiv 0 \pmod{4}, ξ4=1\xi_4= 1, whereas ξ6=1\xi_6 = -1 since 62(mod4)6 \equiv 2 \pmod{4}. Note that ξ0=ξ1=1\xi_0= \xi_1= 1.

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

  • You might use a scientific calculator for this problem.

limx0(cosx)sinx1x3x6= ? \large\displaystyle \lim_{x \to 0} \frac {(\cos x)^{\sin x} - \sqrt{1-x^3}}{x^6} =\ ?

Find f(2016)(0)2016!\dfrac{f^{(2016)}(0)}{2016!} for f(x)=x3(1x)3(1+x+x2)f(x)=\dfrac{x^3}{(1-x)^3(1+x+x^2)}.

Clarifications:

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

  • !! denotes the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

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