Let be any real number and define

for integers . Find the limit

**Incomplete.**

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Stumbling Robot

A Fraction of a Dot
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Find the limit of an expression involving *s*_{n}(a) = 1^{a} + … + n^{a}

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Let be any real number and define

for integers . Find the limit

**Incomplete.**

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.

See exercise 13 (c)- section I 4.10

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

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.

could you please elaborate

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

If a != -1

Can you please explain further?