Replace \(x \) by \( 1-x \), right

Calculus Level 5

For integers, \(\displaystyle n \) and \(\displaystyle k \), related as \(\displaystyle 0 \leq k \leq n ; n \in \mathbb{N}\), a sequence is defined as follows:

\[ I_n (k) = \int \limits_0^1 x^k (1-x)^{n-k} \text{d}x \]

Evaluate: \[ \displaystyle \frac{ \sum_{k=0}^{2014} k I_{2014} (k) }{ \sum_{k=0}^{2014} I_{2014} (k) } \]

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