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1) Пусть n=2
![1+ frac{1}{ sqrt{2} } textgreater sqrt{2} \ \ sqrt{2}* (1+ frac{1}{ sqrt{2} } ) textgreater sqrt{2} *sqrt{2} \ \ sqrt{2} +1 textgreater 2 \ \ sqrt{2} textgreater 1 \ \](https://tex.z-dn.net/?f=1%2B+frac%7B1%7D%7B+sqrt%7B2%7D+%7D+++textgreater+++sqrt%7B2%7D++%5C++%5C++sqrt%7B2%7D%2A+%281%2B+frac%7B1%7D%7B+sqrt%7B2%7D+%7D+%29++textgreater++++sqrt%7B2%7D+%2Asqrt%7B2%7D++%5C++%5C++sqrt%7B2%7D+%2B1+textgreater++2+%5C++%5C+sqrt%7B2%7D++textgreater++1+%5C++%5C "1+ frac{1}{ sqrt{2} } textgreater sqrt{2} \ \ sqrt{2}* (1+ frac{1}{ sqrt{2} } ) textgreater sqrt{2} *sqrt{2} \ \ sqrt{2} +1 textgreater 2 \ \ sqrt{2} textgreater 1 \ \")
верно
2)Пусть верно при n=k
![1+ frac{1}{ sqrt{2} } + frac{1}{ sqrt{3} } +...+ frac{1}{ sqrt{k} } textgreater sqrt{k} \ \](https://tex.z-dn.net/?f=1%2B+frac%7B1%7D%7B+sqrt%7B2%7D+%7D+%2B+frac%7B1%7D%7B+sqrt%7B3%7D+%7D+%2B...%2B+frac%7B1%7D%7B+sqrt%7Bk%7D+%7D++textgreater+++sqrt%7Bk%7D++%5C++%5C+ "1+ frac{1}{ sqrt{2} } + frac{1}{ sqrt{3} } +...+ frac{1}{ sqrt{k} } textgreater sqrt{k} \ \")
3)докажем, что верно при n=k+1![1+ frac{1}{ sqrt{2} } + frac{1}{ sqrt{3} } +...+ frac{1}{ sqrt{k} } + frac{1}{ sqrt{k+1} } textgreater sqrt{k}+ frac{1}{ sqrt{k+1} } \ \](https://tex.z-dn.net/?f=1%2B+frac%7B1%7D%7B+sqrt%7B2%7D+%7D+%2B+frac%7B1%7D%7B+sqrt%7B3%7D+%7D+%2B...%2B+frac%7B1%7D%7B+sqrt%7Bk%7D+%7D+%2B+frac%7B1%7D%7B+sqrt%7Bk%2B1%7D+%7D++textgreater++sqrt%7Bk%7D%2B+frac%7B1%7D%7B+sqrt%7Bk%2B1%7D+%7D++%5C++%5C+ "1+ frac{1}{ sqrt{2} } + frac{1}{ sqrt{3} } +...+ frac{1}{ sqrt{k} } + frac{1}{ sqrt{k+1} } textgreater sqrt{k}+ frac{1}{ sqrt{k+1} } \ \")
положительное число
![sqrt{k} + frac{1}{ sqrt{k+1} } textgreater sqrt{k+1} \ \ sqrt{k+1}*( sqrt{k} + frac{1}{ sqrt{k+1} } ) textgreater sqrt{k+1} * sqrt{k+1} \ \ sqrt{k(k+1)} +1 textgreater k+1 \ \ sqrt{k^2+k} textgreater sqrt{k^2} ;k geq 2 \ \](https://tex.z-dn.net/?f=+sqrt%7Bk%7D+%2B+frac%7B1%7D%7B+sqrt%7Bk%2B1%7D+%7D++textgreater+++sqrt%7Bk%2B1%7D++%5C++%5C+sqrt%7Bk%2B1%7D%2A%28+sqrt%7Bk%7D+%2B+frac%7B1%7D%7B+sqrt%7Bk%2B1%7D+%7D+%29+textgreater+++sqrt%7Bk%2B1%7D+%2A+sqrt%7Bk%2B1%7D++%5C++%5C++sqrt%7Bk%28k%2B1%29%7D+%2B1+textgreater++k%2B1+%5C++%5C++sqrt%7Bk%5E2%2Bk%7D++textgreater+++sqrt%7Bk%5E2%7D+%3Bk+geq+2+%5C++%5C+ "sqrt{k} + frac{1}{ sqrt{k+1} } textgreater sqrt{k+1} \ \ sqrt{k+1}*( sqrt{k} + frac{1}{ sqrt{k+1} } ) textgreater sqrt{k+1} * sqrt{k+1} \ \ sqrt{k(k+1)} +1 textgreater k+1 \ \ sqrt{k^2+k} textgreater sqrt{k^2} ;k geq 2 \ \")
верно
⇒
![1+ frac{1}{ sqrt{2} } + frac{1}{ sqrt{3} } +...+ frac{1}{ sqrt{k} } + frac{1}{ sqrt{k+1} } textgreater sqrt{k+1} } \ \](https://tex.z-dn.net/?f=1%2B+frac%7B1%7D%7B+sqrt%7B2%7D+%7D+%2B+frac%7B1%7D%7B+sqrt%7B3%7D+%7D+%2B...%2B+frac%7B1%7D%7B+sqrt%7Bk%7D+%7D+%2B+frac%7B1%7D%7B+sqrt%7Bk%2B1%7D+%7D++textgreater++sqrt%7Bk%2B1%7D+%7D+%5C+%5C+ "1+ frac{1}{ sqrt{2} } + frac{1}{ sqrt{3} } +...+ frac{1}{ sqrt{k} } + frac{1}{ sqrt{k+1} } textgreater sqrt{k+1} } \ \")
ч.т.д.
верно
2)Пусть верно при n=k
3)докажем, что верно при n=k+1
верно
⇒
ч.т.д.
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