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.