Hi, I do not understand why a limit of a sequence has a curious result. Please, can someone make me clear. The limit follows

In[1]:= Limit[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]
\*SuperscriptBox[\(k\), \(p\)]\)/n^p - n/(p + 1), n -> \[Infinity], 
 Assumptions -> 1 <= p && p \[Element] Integers]

Out[1]= Limit[-(n/(1 + p)) + n^-p HarmonicNumber[n, -p], 
 n -> \[Infinity], Assumptions -> 1 <= p && p \[Element] Integers]

The result is known, that is 1/(p+1). If we change the Assumptions, then we get the correct result.

In[2]:= Limit[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]
\*SuperscriptBox[\(k\), \(p\)]\)/n^p - n/(p + 1), n -> \[Infinity], 
 Assumptions -> 1 <= p]

Out[2]= 1/2

Normally the first limit is calculated by the Stolz-Cesaro theorem followed by the Newton binomial formula. In this case we have no concrete answer. Instead we have in the second case. Cloudy.