利用柯西准则,证明以下各数列的收敛性:[br][/br] 对于数列 [tex=6.429x1.357]cBjbtS+pT+EWSc/PKV+Fk2OTvWXBU30fSKkUCor+Qv8=[/tex], 若存在数 [tex=0.5x0.786]hycNLgozeED/VkKdun7zdA==[/tex], 使得[br][/br][tex=23.857x1.357]WiudHel+CKjlqtaAVhIMg2eC15j9zNbVWaU/oYLoAXSZPAFpcbBRItx10o+t6fEjWRvWgiHowdHw0jQSAuca3gx1Yo0no3zebxrZE8atwJcg0FhRQxML+BmW9chsmbtLyX3NjVlgxYR2lFLZ7aHYwLHZ6osWp+em5CPdb1VVh1U=[/tex][br][/br]则称数列[tex=6.429x1.357]cBjbtS+pT+EWSc/PKV+Fk2OTvWXBU30fSKkUCor+Qv8=[/tex] 有有界变差.[br][/br]证明:凡有有界变差的数列是收敛的.举出一个收敛数列而无有界变差的例子.
证 设[tex=23.857x1.357]AaaHhRAvpwDSwsgX7SHdzJ5fC5Slh9B4dN8KfoVMTRh0MO3/NkFbm+u2asBTveO4fNo4FQ/4qHjXqUCVgrMFw4KYxtHcl7VWWFB9isGTOBPENBcSk7RliYHuCy9ES0PYDCiLThvdOKY7c/d32XgM3kpT1pC1GSefBwYF1ShCmSE=[/tex], 则数列 [tex=1.929x1.357]tdGwsL18+GU5A1tMS4RJiBMRlVXdicshMce1ZE6G3ic=[/tex] 单调增加且有界，[br][/br]所以,它是收敛的.[br][/br] 根据柯西收敘准则，对于任给的 [tex=2.357x1.071]zaTYmiB02c3fW3zvAQdizg==[/tex],存在数 [tex=0.857x1.0]+NBI8Pm2vVS+bGgOpHKyOA==[/tex], 使当 [tex=5.0x1.071]bQ4vgXlNTvYBPv+ONt+r7A==[/tex] 时,[tex=5.214x1.357]MFE9YXJzP4jcIsQvf9BGtT8McsCZkx9zHC6BGMnLrS/H+C6eErIHk06Gh7617d7W[/tex], 即[br][/br] [tex=20.786x1.357]0Ah0627f/HdnbR0YV4y12ckHDPk5ttN5xsbHRCTNtrymyDJadG50ogFosXRNlRVwd8hhtE4cL6HloYpqs3n6sk3yUNuLB3YoGLSlHZz7eayzuWeg6rKosvMAAyh56jJ8WIB8hdl6kbUb8Q+ShmZwrw==[/tex][br][/br] 而对于数列 [tex=2.0x1.357]CjCvAldACdhCbOUJYZLY+0nRBLhCQFA+2tCS8je5CxI=[/tex] 有[br][/br][tex=25.571x3.071]qeiYnKXLEhyhuGRg8yLtr2dk2Kx8MguQtTjU9IotS5aJaZjYQqY/AmNJnWV0XH/rRoWwQkipRm/ZH7oaEa5Fo68KwMnt5+fjT24DtvXhnTn57FrXsxOcAlwHbehZHBU1LKkDaf3t/ftrRPUmRWNGdj5VwxsfKiVjMHKoqqF96e2GitKbPyKDdXx/fn/dfSPtl/2BSkNDNBKsEYIK8DR0RsHVFFkADFpOwoOS05S5cGpl640MNvlGvdS36QZjjEcBzViacuxNTx3BRMpZDUpvsrYPZJ+ANYmrfEQtxn01u86ZB+yGO2KGderGFQ04kPsUfAWmjm0PPwhRR4AZCgldTw==[/tex][br][/br]所以,数列 [tex=2.0x1.357]CjCvAldACdhCbOUJYZLY+0nRBLhCQFA+2tCS8je5CxI=[/tex] 是收敛的. [br][/br] 数列 [tex=19.429x2.357]6s3MZtEzo88tz5QpUxsXSf/gPa9oDIp3mPMeMgor5rPbOuIYV/nZG3hStNd6B7wx8aAIm52vvic0xCO2/R7wTcs4pbM4c2uUupYPiZY8BnJvckNZ/Tk7emtI3KQ5wdN4+auqS0V/YY/Se4eJErRvNw==[/tex], 它是以零为极限的收敛数列. 但它不是有有界变差的. 事实上，[br][/br][tex=21.214x5.929]qeiYnKXLEhyhuGRg8yLtr2PqjY5ba2j092MCOCxXFPwW/u3SbUYLGhRghmh//ozIykHrB1f3BnOUhfNEiHtyhgH5bbnWJijmdQiJaGUSJpbv2xZI8q2MyIxfqOrBc30+HKdoPPuIFehBAGhEoBSVJ8RGHr+tmVdyM1618TVTEj3qof4kfq66qh98wdKdNFTJJkEnmCpFuK0TVEo31vjL0jondVrl755PZqEteW7OzXw2PgysAGlHTLojimbBD6RZxne59OqN0wid9IcwjEsx9JC/cga/LmwO2bl4/Tbb37USkNjVMjCGv+ferBYIPV2CJ7wSVh5a24S4HdjXnHkVNak8PDz5jHCbSF9GAo+uIfylm8/tttqR6yYlZYY1ncHnV1IlXScqPGU8xEOfao1CF2GWXDh4uaT0KvAZ5/KYH7GkZBFDczqH5bqYm06Y3vMU[/tex][br][/br]而数列 $\omega_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$ [tex=21.214x5.929]qeiYnKXLEhyhuGRg8yLtr9oVII0NBxyz0xgcc4PdnrEzkHy4GxIpvIhPlF2KS3gcgjojVcUWfgnb2Ma6M0GPWpZ4ZCp7EMwLDrFnOMctU6QSoYirzl9cQAjv6Nm1VTao2wKE98PbX9ZksEKQaEjhBXArBYNVJQ96r0QIZ6koncJgDM6YmOr+wWwvE7mpYzy+un78KpsrAxZ8FRAOpyY1IXyaCKr31h49WcL7690Kx4ixVvo1VDzgS+yD5sShheV0trduLZ3Ybj2UndW0U+g6wz1xmeaATOtmZM+D//PjWXtRcCgvZBJfSDm+UiEQQRAVD6lKQlG8HSWUyoXJkHZ5ch2z2q5bG0pzzQ3IsU+9ZZqk5f+7NFMyik96Jb+CvUkNFbcHR2mlsXPykqVZAfquPJZDae3OjdcEdKQU4rF6mCcePXfp9Gg71kG/nOXvnyFY[/tex]是发散的, 又是递增的,故 [tex=4.143x1.143]6OqwiZr3+SuKppngCRVhR9wP7nHdetz4m8QeapMeRJXnk7o53blgPP3TWr7QK2Uc[/tex]于是，[br][/br] [tex=15.929x1.357]WiudHel+CKjlqtaAVhIMg2eC15j9zNbVWaU/oYLoAXSZPAFpcbBRItx10o+t6fEjWRvWgiHowdHw0jQSAuca3tTljNk80H4ObZbYbCa7uHqXAaNGYwGWSZPM1XFUO/9q[/tex][br][/br]不是有界的,因而,收敛数列 [tex=10.643x2.357]CjCvAldACdhCbOUJYZLY+1aWukzA4hGiRcug4sZ3043vpl2qBlgu5qeShy0/EOnCTT35PAzPsPqjPw2Ozp3zhQ==[/tex]无有界变差.
举一反三
- 证明 : 从有界的数列[tex=6.429x1.357]cBjbtS+pT+EWSc/PKV+Fk2OTvWXBU30fSKkUCor+Qv8=[/tex]中，永远可选出收敛的子数列 [tex=7.143x1.357]FNqssYTOxeyCYvbDNxfNcAuXVz03Gv7yqk4A/iFTXYA=[/tex]
- 设数列[tex=6.429x1.357]cBjbtS+pT+EWSc/PKV+Fk2OTvWXBU30fSKkUCor+Qv8=[/tex] 收敛，而数列 [tex=6.571x1.357]f1zXik2ypkPVStMgUFRqq1Ji04HbJosZjQRYcpKAOqk=[/tex]发散, 则能否断定关于数列[tex=1.929x1.0]ySN/Gj9N5kGagiN5LMg0mA==[/tex]$x_{n} 的收敛性? 举出适当的例子.
- 设数列[tex=6.429x1.357]cBjbtS+pT+EWSc/PKV+Fk2OTvWXBU30fSKkUCor+Qv8=[/tex] 收敛，而数列 [tex=6.571x1.357]f1zXik2ypkPVStMgUFRqq1Ji04HbJosZjQRYcpKAOqk=[/tex]发散, 则能否断定关于数列 [tex=3.0x1.143]zGBkYiWIVVtqDhc6U2OTjYOkjYi/Z7oK40CbOO1LMcg=[/tex]
- 利用关于单调有界数列极限存在的定理,证明以下各数列的收敛性:[br][/br][tex=15.143x2.786]ULAugtjOYQK+tBrizs5LR1hjz+QlTSiL6Q4HVs4m0iYT1lzn62NtJKwEECIQqv2NayAHEIkwqGeGDh8aa9oJGchWojNPyjphBGLOGS+zMbWBZHJ7eAggeERA+QGilhQ4[/tex]
- 利用关于单调有界数列极限存在的定理,证明以下各数列的收敛性:[br][/br][tex=10.214x2.429]d16fxmCSV3OgJaGsGHuHu0agK6ZmZfSuobd12gSm6R4/8XXYZWlc9x9Ai4ksfk5E4HMbKP10Z+WBcaJDm0rCj+jLpCkzR0uomF+0ylfzGZk=[/tex]
内容
- 0
试用定义1证明：(1) 数列[tex=2.357x2.786]YfOpfsXU462gw1PTp8mUhnyox9SPD+a41vcy562G3qQ=[/tex]不以 1 为极限[br][/br](2) 数列[tex=3.643x2.786]hn0grm0nFoK1+B+qN90oORmIO38IPAPRNvMSoSSLroY=[/tex]发散.
- 1
利用关于单调有界数列极限存在的定理,证明以下各数列的收敛性:[br][/br][tex=15.143x2.786]F5UuQzGR/vAk8AwE/oxMpF41/0Lw3PjwOiPrKK8GZGJJZdSsWp1kja2axERlfQkTo5ttBhB565aGAXx6r8A1u+vhlnVjxYmpR+Ju0LLl1544WY1pnRqZY+ngKWkT/wWf[/tex]
- 2
利用柯西准则,证明以下各数列的收敛性:[br][/br][tex=9.071x2.357]Myo+LsQucIPkoBjnlLnBnp+9thoUR8/CNXmj6UT/5PoHyn81ptZpi10q4QsE3vCO[/tex]
- 3
利用单调有界数列收敛准则证明下列数列的极限存在.[tex=10.143x2.429]PQFiji/X+PAXK5Mf5O9sysjL7nxlk8iGb2TkUn4RS04/yFW9ARVojzc5JrGVjglG[/tex].
- 4
利用 [tex=3.286x1.214]K1reZ+1xSIWCJSAffGD3BQ==[/tex] 收敛原理证明: 单调有界数列必定收敛.