What's Your Approach

Calculus Level 5

Define

f(x)=12tan(x2)+14tan(x4)+18tan(x8)+.f(x) = \dfrac{1}{2} \tan\left(\dfrac{x}{2}\right) + \dfrac{1}{4} \tan\left(\dfrac{x}{4}\right) + \dfrac{1}{8} \tan\left(\frac{x}{8}\right) +\cdots.

Then f(3π8)f\left(\dfrac{3\pi}{8}\right) can be written as abπc+de \dfrac{a}{b\pi^{c}} + d - \sqrt{e}

If ab\dfrac{a}{b} is in simplest form i.e. gcd(a,b)=1\gcd(a,b)=1, and a,b,c,d,eNa,b,c,d,e\in\mathbb N, find the value of a+b+c+d+ea + b + c + d + e.

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