Matematică
Bogdanel12345
2015-11-09 10:17:24
Vă rog să mă ajutați !
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calinusu
2015-11-09 14:43:12

[latex]1 extless dfrac{1}{sqrt3+sqrt2}+dfrac{1}{sqrt4+sqrt3}+...+dfrac{1}{sqrt{n+1}+sqrt{n}} extless 3 quad ($rationalizam) \ \ \$1 extless dfrac{(sqrt3-sqrt2)}{(sqrt3+sqrt2)(sqrt3+sqrt2)}+dfrac{(sqrt4-sqrt3)}{(sqrt4+sqrt3)(sqrt4-sqrt3)}+\ +...+dfrac{sqrt{n+1}-sqrt{n}}{(sqrt{n+1}+sqrt{n})(sqrt{n+1}+sqrt{n})} extless 3 \ \ \1 extless dfrac{sqrt3-sqrt2}{sqrt3^2-sqrt2^2}+dfrac{sqrt4-sqrt3}{sqrt4^2-sqrt3^2} +...+dfrac{sqrt{n+1}-sqrt n}{sqrt{n+1}^2-sqrt{n}} extless 3[/latex] [latex]1 extless dfrac{sqrt3-sqrt2}{3-2}+dfrac{sqrt4-sqrt3}{4-3}+...+dfrac{sqrt{n+1}-sqrt n}{n+1-n} extless 3 \ \ \ 1 extless sqrt3-sqrt2+sqrt4-sqrt3+...+sqrt{n+1}-sqrt n extless 3 \ \ \ 1 extless sqrt3-sqrt2+sqrt4-sqrt3+sqrt5-sqrt4+...+\ \ +sqrt {n}-sqrt{n-1}+sqrt{n+1} -sqrt n extless 3 \ \ \ 1 extless -sqrt2+sqrt{n+1} extless 3 Big|+sqrt2 \ \ 1+sqrt2 extless sqrt{n+1} extless 3+sqrt2Big|^2 \ \ 1+2sqrt2+2 extless n+1 extless 9+6sqrt2+2 Big|-1 \ \ 2sqrt2+2 extless n extless 10+6sqrt2,quad (sqrt2 approx 1,41) \ \ 2cdot 1,41+2 extless n extless 10+6cdot 1,41 [/latex] [latex]2,82+2 extless n extless 10+8,46 \ \ 4,82 extless n extless 18,46 \ \ Rightarrow oxed{n in Big{5,6,7,8,9,10,11,12,13,14,15,16,17,18Big}}[/latex]

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