# A limit of a sequence - curious result

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 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.
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