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

Let be any real number and define for integers . Find the limit Incomplete.

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.

2. Andres Tellez says:

See exercise 13 (c)- section I 4.10

• Andres Tellez says:

Is better to see exercise 34(c)- section 10.4. You can compute the limit easily from there.

• 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.

• Anonymous says:

3. Roberto says:

Hint: Sn(a) is asymptotically equivalent to n^(a+1)/(a+1)

• Roberto says:

If a != -1

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