Power of sine!
Calculus
Level
5
\[I(n)= \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sin ^{ 2n }{ x } } \ dx \qquad \qquad X=\sum _{ r=1 }^{ \infty }{ \frac { I(r) }{ { \left( \begin{matrix} 2r \\r \end{matrix} \right) } } } \]
Given the above, find \(\left\lfloor { 10 }^{ 4 }X \right\rfloor \)
Notation:
- \( \displaystyle \binom{n}{m} = \frac{n!}{n! (n-m)!} \) denotes the binomial coefficient.
- \(\lfloor \cdot \rfloor\) denotes the floor function.
This is part of my set Powers of the ordinary.
by
Raghav Vaidyanathan