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Given the $harmonic\;numbers$ $$H_n=\sum_{k=1}^{n}\frac 1 k ,n\!>\!0,H_0=0$$ prove $$\sum_{k=0}^n H_k=(n+1)H_n-n$$ Also in the terminology according to the book A=B would you not agree that the $H_k$ cannot be identified as the terms in a hypergeometric series ? That is at least not a single hypergeometric series.

user158293
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    According to https://math.stackexchange.com/a/2456007/42969, that identity “simply follows from summation by parts.” – I do not understand the second part of the question. – Martin R Dec 16 '21 at 07:45
  • Also here: https://math.stackexchange.com/q/464957/42969. – Martin R Dec 16 '21 at 07:50
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    Perhaps you can clarify the second part: What is “the terminology according to the book A=B”? Why do you think that the $H_k$ are not terms of a hypergeometric series? – Martin R Dec 16 '21 at 08:01
  • As far as the the terminology i must say it is not all that clear to me either but i think mainly it takes the ratio of the n+1 term to the n'th term and requires it to be a rational function of the summation index. I get that as the index approaches infinity it seems to me to be an infinite number of terms in the ratio expression ? Why do think the $H_k$ are terms of a hypergeometric series and what exactly is that series in terms of the somewhat standard $pFq[...]$ notation of the numerator and denominator entries. – user158293 Dec 16 '21 at 11:24

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