设正项级数 [tex=2.643x1.357]txD3QEY3vcweZJuJhQiJqQ==[/tex]收敛 ,证明级数[tex=5.571x1.357]IAsk7sFNw9hFKuiKwVEkUcwQ1mswCRh67nB4qUbaTe3LRdfGdjm6zwnVDhNGrTr+[/tex] 也收敛.
举一反三
- 设级数[tex=2.643x1.357]txD3QEY3vcweZJuJhQiJqQ==[/tex]满足: 加括号后级数 [tex=10.714x2.714]NizruAL2gjMeKBgOmCx3RjoOfnGQHn1ol8BPEePbogWA7qRBlhGty21KbA3xeq0QesDM8qJwiHZ72WQvr5FHeqAY2HP9M00qD2hLGQzFbZc=[/tex]收敛([tex=2.857x1.286]253O8kwK0+/6rEQec46Ubw==[/tex]), 且在同一括号中的 [tex=2.5x1.286]txFzQKZln4PHucXT6w5E1e0JoNClxZ6pGJz2fPJKou4=[/tex], [tex=2.5x1.286]XMRuYpAw22q6TDR5gfEtmX/+kGW5cYeHOlyNt1g8TqU=[/tex], [tex=1.143x1.286]PZ3wc82RrbgX5KwVcyJcmA==[/tex], [tex=2.214x1.286]bJGRxjNhozq3NIwzs18/Hik4kC0rcYG+6nSnqKKk6PY=[/tex]符号相同. 证明[tex=2.643x1.357]txD3QEY3vcweZJuJhQiJqQ==[/tex]亦收敛.
- 对于级数[tex=2.643x1.357]ALXW+wMh1cD+dclWOJzBRQ==[/tex]有[tex=4.857x2.286]i4iYVluDy/FHSULcqzWo+laJO/hglTHJQU80rPwiISGZag5AO4128RasVAPtfniV[/tex],能否断定级数[tex=2.643x1.357]ALXW+wMh1cD+dclWOJzBRQ==[/tex]不绝对收敛,但可能条件收敛?
- 设级数[tex=3.571x2.714]LCs/jzl+nr3KBTJXBn4IiTaMdvoS/p/hGL/Jv9ntegzmzbVBv3v1HeKEgBlLcyLM[/tex]的收敛半径为[tex=7.286x1.357]sTdvH6zX0iZNqILTrUec+Q==[/tex]，证明：级数[tex=7.143x2.714]LCs/jzl+nr3KBTJXBn4IiVrR/a63aRDgwm6Ulx0DCkqQZXUGezi8qQqRicSofTkUWyb3f6mqFTz2twehW0bB7Q==[/tex]的收敛半径为[tex=7.714x2.786]88n1NtKriG0YM72QT5w50ARKMC3GTCC7OGxgHAYlBdhewQuMEfErAwKQ9wpi7IxVHJewFuEn04JodLhBCFDNBA==[/tex].
- 设正项级数[tex=2.714x2.714]ySadpvq7BrEZCGdcnD6+aevQSiUapl3qI6D/1lvgJVQ=[/tex]收敛，则下列级数中，一定收敛的是[input=type:blank,size:6][/input] . 未知类型：{'options': ['[tex=5.286x2.714]ySadpvq7BrEZCGdcnD6+aUSs8NjqGM34Pj0L/aWjPf5nwpc8Gxc46klh+uULINYq[/tex][tex=4.786x1.286]0IqZGXsQumQ8HVtPWCATrtiJCBHUv8/DIKQIwjftTCI=[/tex]', '[tex=3.571x2.714]ySadpvq7BrEZCGdcnD6+aSrhkX13cc9+5wstZa/wdPBrtZqNaznOw9czyCs260T4[/tex]', '[tex=2.929x2.714]ySadpvq7BrEZCGdcnD6+aXdjt7olXa1sOQ/uQAnHb9YfMOOvt9w4M+ygFFPEXpM3[/tex]', '[tex=5.071x2.714]ySadpvq7BrEZCGdcnD6+aXJu218uJmjxWnvA5oEiEDTma8mLHGCbClj8X9wIEyr2[/tex]'], 'type': 102}
- 试明：若级数[tex=4.429x2.714]ySadpvq7BrEZCGdcnD6+aYYKMve96JE4sd3o4QbcQhyFEB8wySY9WvLLExFxYbd4[/tex]及[tex=4.0x2.714]ySadpvq7BrEZCGdcnD6+aZmrAzqw84m41cRzSztSWcBTzzE1dPl749vdarS2YDQV[/tex]皆收敛且[tex=11.071x1.286]KEk9vvqmh+CBNh5nnxa46HqdwXfcZ7FgwC5TRAiXnHCxUdO0AP0u57VybKAorlBU12GvShicsM4yb7ZMMzYXHg==[/tex]，则级数[tex=4.0x2.714]ySadpvq7BrEZCGdcnD6+aeoIzvM7G7HHNcjbk992lZRvUS0Xh/lotVqiUMKn/3kW[/tex]也收敛，若级数 [tex=1.571x1.357]hjG9g3d/QK7lH8zZx+2L/Q==[/tex]与[tex=1.5x1.357]QON3ZX8tDpa8hcC8KI55XA==[/tex]皆发散，问级数[tex=1.5x1.357]jW5HUp91iXA6ZnNyC/+CoQ==[/tex]的收敛性若何？提示：(1)先证级数[tex=5.5x2.714]ySadpvq7BrEZCGdcnD6+aRrMeIUlIOH1QB+K47pqlk3PJyHm4zPMgvDT8ndQ74qKuwSRORSNDNzjOlQ26lPWRw==[/tex]收敛。(2)可能收敛，也可能发散。例如[tex=11.643x2.357]ozp+yD1eymb4/EK2UAZZC9r19Jhg5MU9Rjw7GnxmzJSPaUrcZlSER0YjHa9TokIA11573J3i6NzMPGTxnbTueQ==[/tex][tex=5.643x1.357]NGWbQ9xjBbPjETUZQumFRw/fmak//iRLrHzvWXLffDA=[/tex]。