Well I found this equation in my old notes and I remember I proved in a scratch paper but I forget it how. Where $H_x$ is the harmonic series til $x$. I remember there is trick to expand $\frac{1-t^x}{1-t}$ to a sum in someway then bring the integral inside and then we will get the harmonic series.
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Please define $H_x$ – Vincent Batens Feb 09 '24 at 22:29
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Is the $x$ integer? – Khosrotash Feb 09 '24 at 22:30
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Connected : https://math.stackexchange.com/a/3641389/305862 – Jean Marie Feb 09 '24 at 22:32
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@VincentBatens I edited it. Should I put in the title, too?? – Mina Basilious Feb 09 '24 at 22:33
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1@Gary Ty. I got it now. Should I delete the question 'cause it is a kind of duplicate? – Mina Basilious Feb 09 '24 at 22:37
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1yes, it is also already flagged as a duplicate and will be probably closed soon anyway. – Vincent Batens Feb 09 '24 at 22:38
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Yes, good idea. – Gary Feb 09 '24 at 22:38
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@Gary Not really a duplicate : here $x$ is a real number, whereas in the mentionned question it is an integer. – Jean Marie Feb 09 '24 at 23:34
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1@JeanMarie I think its an integer since $H_x$ is defined as the harmonic series till $x$. – Vincent Batens Feb 10 '24 at 00:57