f-t-t-0-e-st-tanh-t-dt-lim-t-0-f-t-g-t-1-t-f-t-lim-n-g-n-h-1-n-1-h-

Question Number 205487 by MathedUp last updated on 22/Mar/24

f(t)=t∙∫_0 ^∞ e^(−st+tanh(t)) dt lim_(t→0) f(t)=??? g(t)=(1/t)f(t) lim_(n→∞) {g(n)−Σ_(h=1) ^n (1/h)}=???

$${f}\left({t}\right)={t}\centerdot\int_{\mathrm{0}} ^{\infty} \:{e}^{−{st}+\mathrm{tanh}\left({t}\right)} \mathrm{d}{t} \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({t}\right)=??? \\ $$$$\mathrm{g}\left({t}\right)=\frac{\mathrm{1}}{{t}}{f}\left({t}\right) \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left\{\mathrm{g}\left({n}\right)−\underset{{h}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{{h}}\right\}=??? \\ $$

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