Let the sum of the following infinite sequence be \(N\). \[1,\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{6},\dfrac{1}{4},\dfrac{1}{12},\dfrac{1}{8},\dfrac{1}{20},\dfrac{1}{16},\dfrac{1}{30},\dfrac{1}{32},\dfrac{1}{42} \ldots \ . \] Find the \(\textbf{remainder}\) when \(\displaystyle \Bigl(N^{2014}+1\Bigr)\) is divided by \(\color{Purple}{\textbf{11}}\).

**Hint**:- Is it really \(\textbf{a}\) sequence?

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