Pascal-ish Triangle

Calculus Level 3

Let Sn=11n+1+122n+1+12+133n+S_n=\dfrac{1}{1^n}+\dfrac{1+\frac{1}{2}}{2^n}+\dfrac{1+\frac{1}{2}+\frac{1}{3}}{3^n}+\cdots.

Then, for positive even numbers mm, there is a beautiful relationship between SmS_{m} and the Riemann zeta function ζ()\zeta(\cdot): S2=2ζ(3)S4=3ζ(5)ζ(2)ζ(3)S6=4ζ(7)ζ(2)ζ(5)ζ(3)ζ(4)S8=5ζ(9)ζ(2)ζ(7)ζ(3)ζ(6)ζ(4)ζ(5)Sm=m+22ζ(m+1)k=2m2ζ(k)ζ(mk). \begin{array} { l r c } S_2 &=& 2\zeta(3) \\ S_4 &=& {3\zeta(5)} \quad {-\zeta(2)\zeta(3)} \\ S_6 &=& {4\zeta(7)}\quad {-\zeta(2)\zeta(5)} \quad {-\zeta(3)\zeta(4)} \\ S_8 &=& {5\zeta(9)}\quad {-\zeta(2)\zeta(7)} \quad {-\zeta(3)\zeta(6)} \quad {-\zeta(4)\zeta(5)} \\ & \vdots & \\ S_{m} &=& \displaystyle \frac{m+2}2 \zeta(m+1) - \sum_{k=2}^{\frac m2} \zeta(k) \zeta(m-k). \end{array} However, there is also a relationship between positive odd numbers mm and the Riemann zeta function. Find this relationship and submit your answer as π4S3. \dfrac{\pi^4}{S_3}.


Bonus: Prove the pattern shown above.

×

Problem Loading...

Note Loading...

Set Loading...