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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.

6 comments

      • Anonymous says:

        Others pointed out the solutions for a >= 0 but for a < 0 it's still not clear. However, for a 0 So the whole limit is goes to 1/nC_1/C_2 which goes to 0. The interval -2 to 0 still remains.

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