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Find the limit of an expression involving sn(a) = 1a + … + na

Let a \in \mathbb{R} be any real number and define

    \[ s_n (a) = 1^a + 2^a + \cdots + n^a \]

for integers n. Find the limit

    \[ \lim_{n \to \infty} \frac{s_n (a+1)}{ns_n (a)}. \]


Incomplete.

8 comments

  1. S says:

    To prove the result for arbitrary real a>-1 (for integer a, it’s easier to prove it using induction), by using an approach similar to the one used for proving the theorem 10.11, I proved first that lim_{n -> infinity} [ sum(k^a) / (n+1) ] = 1/(a+1). With that, it’s somewhat obvious how to to derive the final result.

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