Self replicating limit

Calculus Level pending

\[\large {\lim_{n\to\infty} \sum _{ k=1 }^{ n}{ \frac { { k }^{ \frac { 1 }{ a } }\left[ { { n }^{ a-\frac { 1 }{ a } }+ }{ { k }^{ a-\frac { 1 }{ a } } } \right] }{ { n }^{ a+1 } } } } = \ ?\]

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