Question Number 30484 by abdo imad last updated on 22/Feb/18
![1) prove that if f is decreasing function we have ∫_n ^(n+1) f(t)dt <f(n)< ∫_(n−1) ^n f(t) dt . 2) let put S_n = Σ_(k=1) ^n^2 (1/(2(√k))) .calculate [S_n ].](https://www.tinkutara.com/question/Q30484.png)
$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{if}\:{f}\:{is}\:{decreasing}\:{function}\:{we}\:{have} \\ $$$$\:\int_{{n}} ^{{n}+\mathrm{1}} {f}\left({t}\right){dt}\:<{f}\left({n}\right)<\:\int_{{n}−\mathrm{1}} ^{{n}} \:{f}\left({t}\right)\:{dt}\:\:. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{put}\:\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}^{\mathrm{2}} } \:\:\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{{k}}}\:.{calculate}\:\left[{S}_{{n}} \right]. \\ $$