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Question Number 207789 by Ghisom last updated on 26/May/24

$$\mathrm{\forall }r\in \mathbb{R}:H_r=\underset{0}{\stackrel{1}{\int }}\frac{t^r-1}{t-1}dt$$$$H_{r+2}-H_r=1$$$$r=?$$

Answered by MM42 last updated on 27/May/24

$$H_{r+2}-H_r={\int }_0^1\frac{t^{r+2}-t^r}{t-1}dt$$$${={\int }_0^1(t^{r+1}+t^r)dt=(\frac{t^{r+2}}{r+2}+\frac{t^{r+1}}{r+1})]}_0^1$$$$=\frac{1}{r+2}+\frac{1}{r+1}=\frac{2r+3}{r^2+3r+2}=1$$$$\Rightarrow r^2+r-1=0\Rightarrow r=\frac{-1+\sqrt{5}}{2}✓$$$$$$

Commented by Ghisom last updated on 26/May/24

$$\mathrm{thank}\mathrm{you}\mathrm{but}\mathrm{I}\mathrm{think}\mathrm{only}\mathrm{the}\mathrm{positive}$$$$\mathrm{root}\mathrm{is}\mathrm{valid}$$

Commented by mr W last updated on 27/May/24

$$whyisthenegativerootnotvalid?$$

Commented by MM42 last updated on 27/May/24

$$ok\underset{―}{\underbrace{⋚}}$$

Commented by Frix last updated on 27/May/24

$$\underset{0}{\stackrel{1}{\int }}\frac{t^{-\frac{1+\sqrt{5}}{2}}-1}{t-1}dt{\mathrm{doesn}}^{\prime }\mathrm{t}\mathrm{converge}.$$

Commented by Ghisom last updated on 27/May/24

$$\mathrm{yes}$$

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